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 Title:
 Prime ideals in rings of continuous functions
 Creator:
 Kimber, Chawne Monique, 1971
 Publication Date:
 1999
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 English
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 v, 113 leaves : ; 29 cm.
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 Subjects / Keywords:
 Completely regular spaces ( jstor )
Conceptual lattices ( jstor ) Continuous functions ( jstor ) Integers ( jstor ) Mathematics ( jstor ) Root systems ( jstor ) Semigroups ( jstor ) Topological spaces ( jstor ) Topological theorems ( jstor ) Topology ( jstor ) Dissertations, Academic  Mathematics  UF ( lcsh ) Mathematics thesis, Ph.D ( lcsh )
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 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph.D.)University of Florida, 1999.
 Bibliography:
 Includes bibliographical references (leaves 110112).
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 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Chawne Monique Kimber.
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 University of Florida
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 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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Full Text 
PRIME IDEALS IN RINGS OF CONTINUOUS FUNCTIONS
By
CHAWNE MONIQUE KIMBER
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1999
This work is dedicated to those women who preceded me and to those who
are yet to follow.
ACKNOWLEDGMENTS
First and foremost, I express my wholehearted gratitude to my advisor, Jorge
Martinez. In the past few years, his humane guidance has helped me to achieve so
very much, in fact, more than I would ever have hoped. I follow his example both in
becoming a mathematician and a caring teacher, and in the enjoyment of the finer
things in life like wine, cheese, and chocolate.
Also, sincere thanks go to my committee members: Richard Crew, for show
ing me some algebra; Alexander Dranishnikov, for teaching me a heap of topology;
Scott McCullough, for introducing me to real analysis (back when we were both
much younger); and Mildred HillLubin, for expanding my worldview through lit
erature.
Cheers and warm hugs to my friends, neighbors, and family, especially to the
immediate: Johnnie, Charles, Frances, Maribell, Chinene, Jean, and the inimitable
Poopygirl, Cei.
TABLE OF CONTENTS
ACKNOWLEDGMENTS ................................. iii
ABSTRACT .................................... v
CHAPTERS
1 PRELIMINARIES ................... ............ 1
1.1 History ................... ............. 1
1.2 LatticeOrdered Groups .. ..................... 3
1.3 fRings .. .. .. .. ... .. .... .. ... .. .. .. 7
1.4 Rings of Continuous Functions ................. ... 10
1.5 Approaches ..... ............ ............ 15
2 CHARACTERS .................. ............... 18
2.1 Hahn Groups ............................. 18
2.2 Lex Kernels and Ramification ................... 20
2.3 Rank . .. . . . . . .. 26
2.4 Rank via Z#Irreducible Surjections ............... 32
2.5 Prime Character .............................. 36
2.6 Filet Character ....................... ..... 43
3 GENERALIZED SEMIGROUP RINGS .................... 46
3.1 Specially Multiplicative fRings ................. 46
3.2 rSystems and lSystems ...................... 52
3.3 fSystems ................................ 58
3.4 Survaluation Ring and nthRoot Closed Conditions ........ 66
4 RAMIFIED PRIME IDEALS .......................... 75
4.1 Ramified Points .................... ....... .. 75
4.2 Ramified Gapoints .. ..................... 79
4.3 Ramification via CEmbedded Subspaces ............ 84
5 tmQUASINORMAL fRINGS .......................... 88
5.1 Definitions ................................. 88
5.2 (B, m)Boundary Conditions ................... 94
5.3 #X, mQuasinormal and SV Conditions .............. 107
REFERENCES ................................... 110
BIOGRAPHICAL SKETCH .............................. 113
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
PRIME IDEALS IN RINGS OF CONTINUOUS FUNCTIONS
By
Chawne Monique Kimber
May 1999
Chairman: Jorge Martinez
Major Department: Mathematics
Given a completely regular topological space X, we wish to determine the
order structure of < Spec(C(X)), C>, the root system of prime ideals of the ring of
realvalued continuous functions on X; and vice versa.
We present four approaches which give partial solutions to these problems.
First, we define three measures on < Spec+(G), C>, the set of prime subgroups of a
latticeordered group, which determine some arithmetic properties of the group, and
vice versa. Second, given any root system, we construct a generalized semigroup
ring R which is a commutative semiprime fring such that < r(%), C>, its root
system of values, is orderisomorphic to the given root system. Then we characterize
those nonisolated G6points whose corresponding maximal ideal is the sum of the
minimal prime ideals it contains. Finally, we characterize those spaces X for which
C(X) has the property that the sum of any m minimal prime ideals is a maximal
ideal or the entire ring.
CHAPTER 1
PRELIMINARIES
The focus of this dissertation is the order structure of < Spec(C(X)), C>, the
spectrum of prime ideals of the ring C(X) of realvalued continuous functions on a
topological space, X. To start, we informally present the history and give motivation
for the discussion herein. We then review some essentials about latticeordered
groups, frings, and rings of continuous functions in detail and then formally indicate
the manner in which this thesis proceeds.
1.1 History
Our history begins with the independent research by Cech and Stone in 1937
(see the papers [Ce] and [St]), in which they describe a compactification 3X of a
topological space X which has the property that every realvalued continuous func
tion on X extends to a continuous function on #X. Further, via #X, they establish
correspondences between the topological structure of X and certain algebraic prop
erties of its ring C(X) of realvalued continuous functions under pointwise addition
and multiplication. For instance, Stone shows that the maximal ideals of the sub
ring C*(X) of bounded functions are in onetoone correspondence with points of
IX. The map p '+ M*P = {f E C*(X) : f(p) = 0} witnesses this correspon
dence and is a homeomorphism of topological spaces when the set of maximal ideals
of the ring is endowed with the hullkernel (Zariski) topology. In particular, this
shows that for compact Hausdorff spaces X, Y, we have that X L" Y if and only if
C(X) 9 C(Y). The next significant result came in 1939 when Gelfand and Kolo
mogoroff proved in [GK] that the maximal ideals of C(X) are exactly those of the
form MP = {f E C(X) : p E d#xZ(f)}, where Z(f) = {p E X : f(p) = 0} and
pe #X.
In the 1950's, Gillman, Henriksen, Jerison and Kohls began a formal in
vestigation of topological/algebraic correspondences of this form. The elementary
techniques and results are recorded in the text Rings of Continuous functions, [GJ].
Concerning prime ideals, in [GJ, 14.3c] we learn that the prime ideals in C(X), con
taining a given prime ideal, form a chain and in [GJ, 4J], it is shown that the
topology on X is closed under countable intersections (i.e., X is a Pspace) if and
only if every prime ideal of C(X) is maximal. More generally, in [GJ, 14.25] we find
that X has the property that every bounded continuous function on set of the form
X \ Z(f), for some f E C(X), extends to a continuous function on X (that is, X is
an Fspace) if and only if every maximal ideal of C(X) contains a unique minimal
prime ideal. Knowing these three facts, we can describe the graph of the prime ideal
spectrum of C(X) in each case, where vertices are prime ideals and edges indicate
setinclusion. In the Fspace situation, the graph is a disjoint set of strands (one
for each point of #X) with no branching; a Pspace yields a graph consisting solely
of vertices (one for each point of #X).
It is from these topological characterizations of the graphical structure of
the spectrum of prime ideals of C(X) that we formulate our questions. Roughly
speaking, we wish to know:
Is it possible to determine the order structure (under inclusion) of the prime
ideal spectrum of the ring of continuous functions of a given topological space?
Conversely, given a graph, is it possible to construct a topological space such
that the given graph, in some sense, determines the structure of the spectrum
of its ring of continuous functions?
It turns out that both of these are rather ambitious pursuits and the ques
tions must be refined before we can approach them. Extending the knowledge of
properties of the prime ideals of C(X), Kohls published a series of papers ([K1],
[K2], and [K3]) in 1957. In [K2], he addresses the properties of chains of prime
ideals of C(X). First he shows that the quotient ring C(X)/P is totally ordered for
every prime ideal P and concludes that the prime ideals of the quotient ring form a
chain. Second, it is demonstrated that if P is nonmaximal, then the chain of prime
ideals in C(X)/P contains an 7rlset (that is, a totally ordered set E such that for
every pair of disjoint countable subsets A, B C E such that A < B, there exists
c E E such that a < c and c < b for every a E A and every b E B). Hence, the chain
of prime ideals contains at least 2Ns primes. We may thus immediately reduce the
second of our questions to only consider those graphs for which each nontrivial edge
passes through an r77set of vertices. The facts presented in the next three sections
show us that the class of graphs to consider can be further reduced.
In this dissertation, we continue to refine the questions and present four
perspectivesranging from the very general to the very specificwhich give partial
results. In order to properly introduce these approaches, we must recall some facts
and constructs which are fundamental to the ensuing investigation.
1.2 LatticeOrdered Groups
Let (L, <) be a partially ordered set. If a, b E L are incomparable, then we
write a (I b. L is totally ordered if any two elements are comparable. We say that
L is a lattice if any two elements a, b E L have a least upper bound and a greatest
lower bound, denoted a V b and a A b, respectively. A lattice L is distributive if
a A (bV c) = (aV b) A (aV c), and dually for all a, b, cE L.
A group (G, +, 0, <) with partial order < is a latticeordered group (hence
forth, group) if it is a lattice and ifg < h implies that c+g < c+h and g+c < h+c
for all c E G. The majority of the groups we consider are abelian, so the additive
notation here is for convenience. It is important to note that any Lgroup is torsion
free [D, 3.5] and its lattice is distributive [D, 3.17]. A (real) vector lattice is an
fgroup G which is also an Rvector space such that rg > 0 for all positive g E G
and for all positive r E RB
By G+ we mean the set of elements g E G such that g > 0. Each element of G
may be written as a difference of elements of G+ : let g+ = g V 0 and g = (g) V 0,
then g = g+ g. This follows from the fact that g+ A g = 0. The absolute value
of an element is given by Igl = g+ + g. In general, we say that a pair of elements
g, h E G are disjoint if g A h = 0. We will write g < h if ng < h for all n E N.
An Ihomomorphism is a group homomorphism that also preserves the lattice
structure. An Isubgroup H of an Igroup G is a subgroup which is also a sublattice
of (G, <). We call an Isubgroup convex if 0 < g h E H implies that g E H. G(S)
denotes the convex Isubgroup of G generated by the set S C G. When S = {g},
we write G(g). In fact, G(g) = {h E G : 3n E N, IhI < nlgl}. In the special
case that G(g) = G, we call g a strong (order) unit. Let ((G) denote the set of
all convex Isubgroups of G ordered by inclusion. This set is a distributive lattice
under the operations of arbitrary intersection and Vie Hi = G(UIeHi), where
{Hi}iAr C C(G) and I is any indexing set; see [D, 7.10] for details.
Let S C G. Then the polar of S is given by
SI = {g E G: IgI A Is = 0 for all s E S}.
If S is a singleton, say S = {g}, then we write g for the polar of S. Such a polar
is called principal. If g = 0, then g is termed a weak (order) unit. Note that for
any S C G, we have that S E C(G) and (S')" = S. Let T(G) represent the
set of polar subgroups of G ordered under setinclusion. Then T(G) 9C (G), but
in general this is not as a sublattice. Under the operations of arbitrary intersection,
I, and V., Hi = (Uie Hi))", where {Hi)3}i, C (G) and I is any indexing set,
we have that 3(G) is a Boolean lattice, by [D, 13.7].
The convex esubgroups of greatest interest to us are the prime subgroups of
G. These are the subgroups H E C(G) for which any one of the following equivalent
conditions is satisfied (see [D, 9.1],[AF, 1.2.10], or [BKW, 2.4.1]):
1. IfgAh=OthengEHor h H.
2. Ifg,h> O and gAhEH then g H or h E H.
3. The right costs of H are totally ordered.
4. The convex Isubgroups of G containing H form a chain.
As suggested by the terminology and the second condition listed above, the
concept of a prime subgroup is related to that of a prime ideal in a ring. The
difference becomes apparent in considering the final equivalent condition which in
dicates that the prime subgroups form a root system. That is, the graph of the
prime subgroups of G, in which nodes indicate prime subgroups and edges repre
sent containment going up, has the property that incomparable elements have no
common lower bound. Illustrating that prime ideals differ from prime subgroups,
we note that the zero subgroup is the only prime subgroup of the totally ordered
group of integers, Z; whereas, the zero ideal and the ideals which are generated by
a prime integer comprise the set of prime ideals of Z. The structure of the graph of
the prime subgroups is the subject of our investigation. Note that we use Spec+ (G)
to denote the set, or spectrum, of all prime subgroups of G and to stand for the
associated graph.
By Zorn's Lemma, minimal prime subgroups exist. Let Min+(G) denote the
set of all minimal prime subgroups of G. If P E Min+(G) then by [AF, 1.2.11] we
have P = U{g : g V P}. This implies 0(Q) de= n{P E Min+(G) : P C Q} is the
set U{g : g V Q}, for a prime subgroup Q C G, by [BKW, 3.4.12].
A basis for an egroup G is a maximal pairwise disjoint set {gi}i4W C G+ such
that for each i E I, the set {g E G+ : g : g,} is totally ordered. The following is
Conrad's Finite Basis Theorem presented as [D, 46.12] and [C, 2.47]. It will figure
in our discussion in the next chapter.
Theorem 1.2.1. Let G be an tgroup. The following are equivalent:
1. G has a finite basis.
2. Min+(G) is finite.
3. q(G) is finite.
4. There is a finite upper bound on the number of pairwise disjoint elements of
G.
5. There is a finite upper bound on the number of elements of strictly increasing
chains of proper polars.
Another application of Zorn's Lemma establishes the existence of convex I
subgroups which are maximal with respect to not containing a fixed element g E G.
Any such subgroup is generally termed a regular subgroup and specifically called
a value of g. The set of all regular subgroups of G is usually represented by r(G).
Regular subgroups are prime, by [D, 10.4], and a prime subgroup is precisely a convex
Isubgroup which is an intersection of a chain of regular subgroups, [D, 10.8]. In
particular, the minimal prime subgroups of G correspond to the maximal chains
in r(G). For these reasons, we call the root system given by P(G) the skeleton of
Spec+(G). By convention, we view F(G) as a partiallyordered indexing set F whose
elements are denoted by lower case Greek letters and then represent the regular
subgroups by V, for 7 E r.
Topologize Spec+ (G) using the hullkernel (or Zariski) topology whose open
base is given by U(g) = {P E Spec+(G) : g V P} for all g E G. In this topology,
Spec+(G) is Hausdorff if and only if Spec+ (G)=Min+(G) by [CM, 1.4]; on the other
hand, in the subspace topology, Min+(G) is always Hausdorff and U(g) n Min+(G)
is both open and closed for every g E G. The space Spec (G) is compact if and only
if G has a strong unit, by [CM, 1.3]; it is demonstrated in [CM, 2.2] that Min+(G)
is compact if and only if G is complemented, that is, if and only if for every g E G+
there is an h E G+ such that g A h = 0 and g V h is a weak unit.
1.3 fRings
Let (R, +, ., <) be a ring whose underlying group is an egroup and satisfies
the relations rc < sc and cr < cs whenever r < s and c > 0. Such a ring is a lattice
ordered ring (abbreviated Iring). If an iring R also satisfies ca A b = ac A b = 0
whenever a A b = 0 and c > 0, then R is called an fring. The following is found in
[BKW, 9.1.2]:
Theorem 1.3.1. Let R be an Iring, then the following are equivalent:
1. R is an fring.
2. Every polar in R is an ideal.
3. Every minimal prime subgroup of R is an ideal.
It is not difficult to verify that every Gring which is 1isomorphic to a subdi
rect product of totallyordered rings (with coordinatewise operations) is an fring.
In [BP], it is shown that the converse of this statement holds when we assume the
Axiom of Choice (abbreviated, AC). Since we routinely apply AC, let us formally
state that we will work within the axioms of ZFC. Then we may use this equiv
alent definition of an fring R in order to obtain this list of arithmetic properties
given in [BKW, 9.1.10], for a, b, c E R:
1. Ifc > 0 then c(aV b) = ca V cb and (aV b)c = acV bc.
2. Ifc > 0 then c(a A b) = ca A cb and (aA b)c = ac A be.
3. IallbI = jabl.
4. If aAb = 0 then ab = 0.
5. a2 > 0.
An fideal of an Iring R is an ideal which is a convex Isubgroup of R. We
call an 1ideal a prime idealif it is also a prime ideal. Let Spec(R) denote the space
of all prime ideals of R in the hullkernel toplogy. Let Max(R) and Min(R) denote
the subspaces of maximal and minimal prime ideals, respectively. By property (4)
above, we see that prime Iideals of an fring are prime subgroups, hence, as in the
case of Igroups, the subset Spect(R) of prime Iideals forms a root system. Denote
the subspaces of maximal and minimal prime Iideals by Maxt(R) and Mint(R),
respectively.
We call a commutative ring semiprime if it contains no nonzero nilpotent
elements. In the case of commutative frings, we have [BKW, 9.3.1]:
Theorem 1.3.2. Let R be a commutative fring, then the following are equivalent:
1. R is semiprime.
2. For any a, b E R, we have that [al A lbI = 0 if and only if ab = 0.
3. Every polar of R is an Iideal which is an intersection of prime ideals.
4. Min(R) = Min(R).
5. R is eisomorphic to a subdirect product of totallyordered integral domains.
We say that an Iring R with multiplicative identity, 1, has the bounded
inversion property if a > 1 implies that a is a multiplicative unit. By [HIJo, 1.1],
a commutative fring R with 1 has the bounded inversion property if and only if
Max*(R) = Max(R).
Let A be any commutative ring. Then, in the hullkernel topology, Min(A) is
a Hausdorff space with a base of clopen sets. If A is a semiprime ring, then Spec(A)
is Hausdorff if and only if Min(A) =Max(A); this occurs if and only if A is von
Neumann regular (or absolutely flat,) i.e., for every a E A there exists b E A such
that a = a2b), see [AM, p. 35]. In [HJ] it is demonstrated that if A is a semiprime
fring, then Min(A) is compact if and only if A is complemented (i.e., for every
a E A there exists b E A such that ab = 0 and a + b is not a zerodivisor). Max(A)
is compact for any commutative ring A with identity, and if A is a commutative
fring with identity which has the bounded inversion property, then the subspace
is Hausdorff, see [HJo, 2.3].
Let A be a commutative ring with identity and P E Spec(A). Define
O(P) = {a E A: 3b P,ab = 0}.
If A is also a semiprime fring, then this is the same as the Isubgroup O(P) defined
in the previous section. Recall that the localization of A at P is the subring, Ap, of
the classical ring of quotients of A/O(P) consisting of the elements whose denomi
nator is not in P/O(P). (For a review of this construction and general facts about
localizations, see [AM] or [G]). It is the case that Ap is a local ring whose unique
maximal ideal is generated by P/O(P) and there is a onetoone correspondence
between prime ideals of Ap and the prime ideals Q of A such that O(P) C Q C P.
Thus, if A is also an fring with bounded inversion, then by the root system struc
ture of Spece(A), we have that AM "I A/O(M) since the quotient ring is already
local with unique maximal ideal M/O(M).
1.4 Rings of Continuous Functions
Let X be a Hausdorff topological space. X is called completely regular (or
Tychonoff) if for every closed set A C X and z E X \ A, there exists a realvalued
continuous function on X such that f(x) = 1 and f(A) = {0}. Unless otherwise
stated, we assume that all spaces are completely regular. Let C(X) denote
the set of realvalued continuous functions on a space X. Under the operations of
pointwise addition and multiplication, C(X) is a semiprime ring. Order the ring
via: f < g if and only if f(x) < g(x) for all x E X. This ordering gives an fring
structure such that C(X) has the bounded inversion property. Let C*(X) denote
the convex Isubring of bounded functions.
The zeroset of f, is the set Z(f) = {x E X : f(x) = 0}. The complement,
coz(f) = X \ Z(f), is the cozeroset of f. By [GJ, 3.6], a Hausdorff space X is
completely regular if and only if its topology is the same as the weak topology
generated by C(X). Equivalently, the set of all zerosets, Z(X), is a base for the
closed sets of such a space, [GJ, 3.2].
Sets A, B C X are completely separated if there exists f E C(X) such that
f(A) = {0} and f(B) = {1}. If for every f E C(A) there exists f E C(X) such that
fAA = f, then we say that A is Cembedded in X. Likewise, A is C*embeddedin X if
bounded continuous functions on A extend to bounded continuous functions on X.
These embedding properties are characterized by complete separation of particular
subsets, as follows:
1. Urysohn Extension Theorem [GJ, 1.17]: A C X is C*embedded in X if and
only if any two completely separated sets in A are completely separated in X.
2. A C*embedded set is Cembedded if and only if it is completely separated
from every zeroset disjoint from it, [GJ, 1.18].
Recall that a Hausdorff topological space X is normal if any two disjoint
closed sets are separated by disjoint open sets. Assuming this stronger separation
axiom, the results listed above give rise to the theorem stated in [GJ, 3D], in which
the equivalence of the first two statements is known as Urysohn's Lemma.
Theorem 1.4.1. Let X be Hausdorff. The following are equivalent:
1. X is normal.
2. Any two disjoint closed sets of X are completely separated.
3. Every closed set of X is C*embedded in X.
4. Every closed set of X is Cembedded in X.
For many reasons, it is often preferable to work with compact spaces. The
StoneCech compactification 3X is our compactification of choice, since fX is char
acterized by the property that it is (up to homeomorphism) the unique compact
space in which X is dense and C*embedded. There are at least three different ways
to construct iX, we begin with the one based on ultrafilters, described in detail in
Chapter 6 of [GJ], which we now summarize.
Let X be a completely regular space, let C be a subset of the power set of
X and Y C C. is a Cfilter if 0 V ., it is closed under finite intersections and if
for every F E 7, the fact that F C F' E C implies that F' e F. If 7 is a Z(X)
filter, then T is also called a zfilter. A maximal filter is an ultrafilter; similarly, a
zultrafilter is a maximal zfilter. Let fX be the set of all zultrafilters on X which
we index by {A' : p E X}. A closed base for the topology on fX is given by sets
of the form Z = {p E XX : Z e AP}, for Z E Z(X). Let p E [X and define
MP = {f E C(X): p E clxZ(f)}.
The theorem of Gelfand and Kolmogoroff [GK] is stated simply as:
Theorem 1.4.2. For a completely regular space X, the set Max(C(X)) is given by
{MP : p E X}.
In fact, this result gives rise to a homeomorphism of fX with Max(C(X)).
That is, since the sets Z[MP] = {Z(f) : f E MP} are precisely the zultrafilters on
X, by [GJ, 2.5], and Theorem 1.4.2 shows that the map p 1+ Z[MP] is the desired
correspondence.
If p E X, then we will write Mp and, in this case, the maximal ideal and
corresponding zultrafilter are called fixed. Otherwise, a maximal ideal and its
corresponding zultrafilter is called free. It is evident that X is compact if and only
if every maximal ideal of C(X) is fixed. Maximal ideals are also classified by the
residue field C(X)/MP. Identifying the constant functions with their constant, we
see that these fields always contain a copy of R We call a maximal ideal real if the
field is exactly R; otherwise, the maximal ideal is called hyperreal. This concept is
the basis for considering the Hewitt realcompactification of X. Denoted vX, it is the
smallest subspace of (X in which X is dense and such that every maximal ideal of
C(vX) is real. In fact, by [GJ, 8.5], vX is the largest subspace of fX in which X
is Cembedded.
With these facts about the maximal ideals firmly in place, we now proceed
to consider the nonmaximal prime ideals. We know that every prime ideal of C(X)
is convex, by [GJ, 5.5]; so we deduce that Spec(C(X)) is a root system. In order
to understand this root system, we are required to consider the properties of other
ideals. For instance, for p E #X, the ideals of the form
0" = O(M0) = {f E C(X) : claxZ(f) is a neighborhood of p}
are of paramount interest when examining the prime ideals of C(X). One reason is
given in [GJ, 7.15]:
Theorem 1.4.3. Every prime ideal P in C(X) contains OP for a unique p E #X
and MP is the unique maximal ideal containing P.
If OP is prime, then we call p an Fpoint. If X has the property that OP is
a prime ideal for every p E X, then we call X an Fspace. We see that in this case,
the graph of Spec(C(X)) consists of a set of strands with no branches. Note [GJ,
14.25]:
Theorem 1.4.4. Let X be completely regular. The following are equivalent:
1. X is an Fspace.
2. /X is an Fspace.
3. The prime ideals contained in any given maximal ideal form a chain.
4. Every cozeroset of X is C*embedded.
5. Any two disjoint cozerosets of X are completely separated.
6. Every ideal of C(X) is convex.
7. Every finitely generated ideal of C(X) is principal (i.e., C(X) is BEzout).
A special case of an Fpoint is when Op = Mp and we call p a Ppoint if
this occurs. Call X a Pspace if every point of X is a Ppoint. In this case, the
spectrum of C(X) consists only of vertices. Equivalent definitions of Pspace are
presented in [GJ, 14.29] and are recorded below. First, recall that an ideal I of C(X)
is called a zideal if f E I and Z(f) = Z(g) implies that g E I. It is immediate
from the definitions that MP and OP are zideals for all p E 8iX. Note that not all
prime ideals are zideals; however, the following says that this is the case in a von
Neumann regular ring.
Theorem 1.4.5. Let X be completely regular. The following are equivalent:
1. X is an Pspace.
2. vX is an Pspace.
3. Every prime ideal of C(X) is maximal.
4. Every cozeroset of X is Cembedded.
5. For each f e C(X), the zeroset Z(f) is open.
6. Every ideal of C(X) is a zideal.
7. For every f e C(X), there exists g E C(X) such that f = gf2 (that is, C(X)
is von Neumann regular).
We now recall the definitions of other types of spaces which are useful to us.
X is basically disconnected if the closure of any cozeroset is clopen. X is extremally
disconnected if any open set has open closure. Discrete spaces are extremally dis
connected; extremally disconnected spaces are basically disconnected and all such
spaces are Fspaces by [GJ, 14N.4]. Every Pspace is basically disconnected by [GJ,
4K.7]. A space is a quasiF space if every dense cozeroset is C*embedded. Clearly,
from [GJ, 14.25], we see that every F space is quasiF. The converses of the pre
ceding statements do not hold. That is, these are distinct classes of spaces, as we
now illustrate.
Example 1.4.6. Consider the following spaces:
1. Let U be a free ultrafilter on N. Let E = N U {a}, in which points of N
are isolated and neighborhoods of a are of the form U U {1}, where U E U.
Then E is an extremally disconnected subspace of #iN, but not a Pspace. In
particular, O is a prime ideal which is not maximal; see [GJ, 4M]. Therefore,
E is an Fspace.
2. Let D be an uncountable set. Let AD = D U {A}, where points of D are
isolated and a neighborhood of A is given by any cocountable set containing
it. Then AD is basically disconnected, but not extremally disconnected by [GJ,
4N.3]. Moreover, the topological sum X = AD II E is basically disconnected,
but neither extremally disconnected nor a Pspace, by [GJ, 4N.4].
3. The corona, fN \ N is a quasiF space which is an Fspace, yet not basically
disconnected; see [GJ, 6W.3, 140].
1.5 Approaches
Starting as generally as possible in Chapter 2, we define three cardinalvalued
characters on the spectrum of prime subgroups of an fgroup. The value of each
measure determines a portion of the arithmetic and/or polar structure of the I
group, and vice versa. For instance, we define the prime character, ir(G) of an
group, G to be the least cardinal K such that for any family {Q0a}a<, Min+(G),
of distinct minimal prime subgroups, we have that Va<, Qa is the smallest convex
Isubgroup of G containing all the elements of Min+(G). Roughly speaking, it is
a measure of the complexity of minimal paths in the graph of Spec+(G) between
minimal prime subgroups. We will show that the measure being finite satisfies the
following, where lex(G) denotes the smallest convex (subgroup of G containing all
the elements of Min+(G) :
Proposition 1.5.1. Let G be an (group and m a positive integer. The following
are equivalent:
1. r(G) = m < oo.
2. m is minimal with respect to the property that lex(G) = G(Uji= a ) for any
m pairwise disjoint positive elements, {aj}>Ti C lex(G)+.
8. m is minimal such that for any prime P C lex(G), the chains of proper polars
in P have length at most m 1.
Chapter 3 is devoted to a discussion of the properties of F(A, R), the gener
alized semigroup ring of realvalued maps on a root system A (which has a partially
defined associative operation, +) each of whose support is the join of finitely many
inversely wellordered sets. The ring structure on this group is introduced in [Cl]
and [C2]; we endow this ring with an fring structure. In particular, we show that
if (A, +) is a root system such that each of the following holds:
1. + is associative (when it makes sense);
2. if a, / E A are comparable, then a + 3, / + a are defined;
3. if a < p and a + 7, / + 7 are defined, then a + 7 < P + 7 and if 7 + a, 7 +
are defined then 7 + a < 7 + 3;
4. and if j is maximal, then 6 + p + 6, p + p are defined and 6 + = ~+6 = 6
for every 6 < IA,
then F(A, R) is an fring if and only if 6 = a + P implies a,3 > 6. And when
this occurs, the fring is semiprime and satisfies the bounded inversion property.
Moreover, by [CHH, 6.1], given any root system A, one of these frings has A
orderisomorphic to its root system of values. Thus, the second of our questions is
answered in the class of frings on the level of skeletons.
However, the solution to the second problem remains unclear in the smaller
class of rings of continuous functions. To gain a modicum of clarity on the situation,
we look to the work of Attilio LeDonne, published in 1977 in [Le], in which he
addresses the incidence of branching in the graph of Spec(C(X)). He shows, for
instance, that the root system branches at every prime zideal when X is a metric
space. In [Le, 2], LeDonne includes a result of DeMarco which states that there
is branching at each Mp when X is a firstcountable space and p is nonisolated.
In Chapter 4, we show that, for a nonisolated Gapoint of a completely regular
space, there is branching at Mp if and only if X \p is not C*embedded in X. This
result is then used to examine branching in Spec(C(X)) when X is not necessarily
firstcountable.
Both of our questions are addressed in Chapter 5, in which we generalize a
few of the results of Suzanne Larson on quasinormal frings that are found in the
series of papers [Lal], [La2], and [La3]. The semiprime commutative quasinormal
frings with identity are the ones having the property that the graph of the root
system of prime ideals does not contain a subgraph of the form:
A (1.1)
By [La3, 3.5], a normal space X has the property that C(X) is quasinormal if
and only if cl(U) n cd(V) is a Pspace for any disjoint cozerosets U, V C X. Our
generalizations similarly describe those normal spaces X for which Spec(C(X)) does
not contain a subgraph of any of the following forms:
a2 an k k k k k (1.2)
where n, k, a1,... aO are positive integers satisfying some specified conditions.
CHAPTER 2
CHARACTERS
We seek a collection of measures on root systems whose values will determine some
portion of the structure of a latticeordered group. In this chapter we describe three
such measures: rank, prime character, and filet character. The rank measures the
width of a connected component of the spectrum, the prime character determines,
roughly speaking, the complexity of minimal paths between minimal primes, and
the filet character counts the maximum length of a chain of branching incidences.
The first sections of this chapter are a review of two constructs essential to the
discussion to follow.
2.1 Hahn Groups
To begin, we recall a method of constructing examples of egroups having a
specified root system as the skeleton of its prime spectrum. Let A be a root system
and define
V(A,R) = {v : A 4 R : supp(v) has ACC},
where supp(v) = 16 E A : v(6) # 0}. V(A, R) is an egroup under pointwise addition
ordered by the relation: v > 0 if and only if v(6) > 0 for every maximal element
6 E supp(v). This fgroup is called a Hahn group. In the paper of Conrad, Harvey
and Holland [CHH], it is demonstrated that any abelian fgroup can be embedded
in a Hahn group of a more general description than we give here. Of interest to
us is the fsubgroup of maps with finite support denoted by E(A, R) and the 
subgroup of maps whose support is the join of finitely many inversely wellordered
sets, denoted by F(A,R). Clearly, E(A,R) C F(A,R).
The proof of the first statement of Proposition 2.1.1 is analogous to that
of Theorem 6.1 in [CHH]. This establishes that r(E(A, R)), r(F(A, R)) and A are
isomorphic as partiallyordered sets. For the sake of completeness, we present an
elementary proof of this fact for the case of F(A, R), although the result is easily
obtained from the theory of finitevalued fgroups. The proof is identical in the case
of E(A, R).
Recall that an group is finitevalued if each element has only a finite number
of values. A special value is a prime subgroup which is the unique value of an element.
An group G is finitevalued if and only if every value of G is special and if and
only if every element of G is a finite sum of pairwise disjoint special elements; for
details, see [AF, 10.10]. If G has a set S of special values such that S is a filter and
AS = {0}, then G is called specialvalued.
Proposition 2.1.1. Let A be a root system. For each 6 E A define
V = {f E F(A, R) : v(7) = 0 when 7 > 6}.
Each V6 is a special value. Further, every value of an element of F(A, R) is of
the form V6 for some 6. Thus, A is the skeleton of Spec+(F(A,R)) and F(A,R) is
finitevalued.
Proof: Let 6 E A and let Xs E F(A, R) be the characteristic function on {6}. Then
Xs 4 Vs and we will show that V6 is the unique value of XS. Let V be a value of Xg
and let v E V+ \ V6. Then there exists 7 > 6 such that 7 is maximal in supp(v), and,
hence, v(7) > 0. If 6 < 7, then 0 < Xs < v, a contradiction. If 6 = 7, then there
exists a positive integer n such that 0 < Xa < nv and hence X; E V by convexity,
which is a contradiction. Thus V = V6.
Let v E F(A,R)+ and let V be a value of v. Let D be the finite set of
maximal elements of supp(v). Then the characteristic function XD is not in V; else,
there exists an integer n such that 0 < v < nXD, a contradiction. Since V is prime
and the set {xa : 6 E D} is pairwise disjoint, there exists a unique element 6 E D
such that Xj 4 V. By the above, we know that V C Vi. Finally, since v V Va, we
have that V = V5, as desired.
The final statement follows from [AF, 10.101 since we have shown that every
value is special. *
2.2 Lex Kernels and Ramification
Throughout, we will describe the location of a prime subgroup in the graph
in reference to a designated convex fsubgroup, called the lex kernel of an group G
and denoted by lex(G). It is the least convex subgroup containing all the minimal
prime subgroups of G. It is always the case that lex(G) is a prime subgroup [D,
27.2] which is normal in G [D, 27.13]. The following is a summary of a part of the
discussion of lex kernels in [D, 27] and gives a description of the subgroup in
terms of its generators.
Proposition 2.2.1. Let G be an tgroup and let C be a convex subgroup. The
following are equivalent:
1. C = lex(G).
2. C is the least prime subgroup such that if 0 < g V C then g > h for every
hE C.
3. C is the convex fsubgroup of G generated by {g E G : g (I 0}.
4. C = {0} U {g E G: 3g1,g2,...g n E G,g II 1 g2 II ." II II 0}.
5. C is the convex Isubgroup of G generated by the nonunits of G.
6. C is prime and is the smallest among all convex tsubgroups of G which are
comparable with every convex tsubgroup of G.
7. C is the maximal convex fsubgroup of G such that lex(C) = C.
8. C is the supremum of the proper polars of G in the lattice of convex fsubgroups
of G.
It is natural to now introduce a concept which we will discuss in more detail in
Chapter 4. This is a generalization of a concept from [Le]. Let A be a commutative
ring with identity and for each a E A, let Max(a) = {M e Max(A) : a E M}. Recall
that an ideal I of A is a zideal if a E I and Max(a) = Max(b) imply that b E I.
Definition 2.2.2. Let A be a commutative fring with identity. A prime tideal P
is ramified if it is the sum of the minimal prime ideals that it contains. A maximal
tideal M is totally ramified if every prime zideal contained in M is ramified. A
completely ramified ring is one in which every prime zideal is ramified.
Graphically, a prime fideal P < A is ramified if and only if it is minimal or
if the root system of prime Gideals of A branches at P. We begin with the fgroup
characterization of ramification. It is the case that a ramified maximal fideal M
of A is the lex kernel of the local fring A/O(M). In order to discuss a proper lex
kernel in an fring, A, we must operate inside a localization. Henceforth, we will
obtain results for local rings and tacitly extend to the general case by referring to
localizations.
The following characterization of ramified maximal fideals is immediate from
Proposition 2.2.1.
Corollary 2.2.3. Let A be a commutative semiprime local fring with identity and
bounded inversion and let M be the maximal ideal. The following are equivalent:
1. M is ramified.
2. M is the convex tsubgroup of A generated by {f E A: f  0}.
s. M= {O}U{f EA:3fi,f2,... fn E A,f II fi f2. l f IIf 0}.
4. M is the convex Isubgroup of A generated by the set
{f E A:3g A,g> O,gA f =0}.
5. M is the convex Isubgroup generated by the elements of A which are not order
units.
6. M is the smallest among all convex esubgroups of A which are comparable
with every convex lsubgroup of A.
7. M is the supremum of the proper polars of A in the lattice of convex fsubgroups
of A.
It is wellknown that the lex kernel of an fgroup is a prime subgroup (see [D,
27.2]). We now show that the lex kernel of a commutative local semiprime fring
with identity is an ideal and then give conditions which guarantee that the lex kernel
is a prime ideal.
Let A be a commutative fring with identity. Recall that an ideal I < A is
pseudoprime if ab = 0 implies a E I or b I. An ideal J < A is semiprime if a E J
whenever a2 E J. A is squareroot closed if for any 0 a E A, there exists 0 < b E A
such that a = b2. Let a, b E A, then A is nconvex if whenever 0 < a < b", there
exists u E A such that a = bu.
Proposition 2.2.4. Let A be a commutative semiprime local fring with identity.
Then lex(A) is a prime subgroup which is a pseudoprime fideal. If, in addition, A
is squareroot closed, then lex(A) is a semiprime fideal.
Proof: Let f E lex(A). Then there exists g > 0 such that f A g = 0. If af = 0
then af E lex(A); else, af A g = 0 and we conclude again that af E lex(A). Hence
lex(A) is an ideal.
Let N be the set of nonunits of A and recall that lex(A) = A(N). Let ab = 0.
If a or b is 0, then there isThen by convexity, we see that a+, a E A(N). Hence
a E A(N) and we have that the lex kernel is pseudoprime. Since any prime ideal is
semiprime and the lex kernel is the sum of the minimal prime ideals, the lex kernel
is semiprime, if A is also squareroot closed; see [HLMW, 2.12(d)]. n
Corollary 2.2.5. Let A be a commutative semiprime local fring with identity and
bounded inversion and let M be the maximal ideal. M is ramified if and only if
lex(A) is a zideal.
Proof: Since the maximal 1ideal is the only zideal of a local fring, this is
immediate. *
Corollary 2.2.6. If A is a commutative local 2convex semiprime fring with iden
tity which is squareroot closed, then the lex kernel of A is a prime ideal.
Proof: By the remark after [La4, 4.2], under these hypotheses, we have that an
ideal is a prime ideal if and only if it is pseudoprime and semiprime. n
For the remainder of this section, let G be an abelian latticeordered group.
Recall the following for H an subgroup of G. H is rigid in G if for every h E H
there is g E G such that h" = g". It is shown in [CM, 2.3] that if H is rigid in G
then the contraction of minimal prime subgroups of G to minimal prime subgroups
of H is a homeomorphism of minimal prime spaces. If H E I(G), then H is very
large in G if it is not contained in any minimal prime subgroup of G. It is shown in
[CM] that if H E (G) then H is very large in G if and only if H is rigid in G. It
turns out that ramification in a rigid subring indicates global ramification and vice
versa. This is a direct consequence of the lex kernel correspondence demonstrated
below.
We will also need the following facts (see [BKW, 2.4.7, 2.5.8]):
Proposition 2.2.7. Let H E C(G).
1. The contraction map from the set of prime esubgroups of G not containing H
to the set of prime Isubgroups of H is an orderpreserving bijection.
2. If V is a value ofh E H in G, then V V n H is a bijection between the set
of values of h in G and the set of values of h in H.
Proposition 2.2.8. Let H < G be a convex isubgroup. Assume that u E H is a
weak unit of H and a weak unit of G. Let V be a value of u in G. Then we have
that V n H = lex(H) if and only if V = lex(G).
Proof: Assume that H C P E Min (G) then u E P and we have that u1D 1 P by
[AF, 1.2.11]. This is a contradiction since u'G = 0 e P. Thus H is rigid in G since
it is a convex Isubgroup which is very large in G.
Assume that V n H is the lex kernel of H. Then V n H is the least convex
Isubgroup of H containing all the minimal prime subgroups of H. Since H is rigid
in G, by the bijection given in the first part of Proposition 2.2.7, all the minimal
prime subgroups of G are contained in V, and V is the least such convex Isubgroup
of G. That is, if W c V also contains the minimal prime subgroups of G, then
W n H is a convex subgroup of H containing all the minimal prime subgroups of
H and hence W n H = V n H. But this says that W n H E F(H) is a value of u
and hence, W E r(G) is a value of u Therefore V = W.
If V is the lex kernel of G then VnH contains all the minimal prime subgroups
of H. Thus the lex kernel of H is contained in V n H. Let P C V f H be a
prime Isubgroup of H containing all the minimal prime subgroups of H. Then by
Proposition 2.2.7, there exists a prime convex Isubgroup Q < G not containing
H such that P = Q n H and since we have a rigid embedding, Q contains all the
minimal prime Isubgroups of G. Hence, Q = V, P = V nH and V n H is the lex
kernel of H. w
Let A be a commutative semiprime fring with identity and bounded in
version. If M E Max(A) then AM is semiprime with bounded inversion. This is
a result of the wellknown facts that the ehomomorphic image of an fring with
bounded inversion has bounded inversion and that AM  A/O(M); see the proof
of [La3, 2.7]. Since we must localize an fring in order to have a proper lex kernel,
the following allows application of Proposition 2.2.8 to frings.
Proposition 2.2.9. Let B be a commutative semiprime fring with identity and
let A be a rigid convex fsubring of B. If M E Max,(A) is such that M = N n A
for some N E Maxe(B), then AM is a rigid convex fsubring of BN.
Proof: Recall that AM c A/O(M) and BN 9 B/O(N). Define a map 4 : A + BN
by a + a+O(N). This map is an fring homomorphism. We show that the image is
convex in BN and that the kernel is O(M). For a E A, let 0 < b+O(N) < a+O(N).
Then there exists n E O(N)+ such that 0 < b < a+n. If bn < 0 then 0 < b < n and
hence b E O(N) since O(N) is convex in B. Thus we may assume that 0 < bn < a.
Then b n E A and b + O(N) = b n + O(N) Elm(n). Therefore the image of 4 is
convex in BN.
The kernel of 4 is O(N) n A. It is easy to show that O(N) nA C O(M) since
M = N n A. For the reverse inclusion, assume that a E A and a O(N). Since
O(N) is the intersection of the minimal prime ideals of B contained in N, there exists
P EMin(B) such that P C N and a V P. By the rigidity of A in B, Pn A EMin(A),
and therefore a V O(M). We now have that Ker(4) = O(N) n A = O(M) and
therefore AM is a convex fsubring of BN.
Since AM contains the identity element of BN, AM is very large in BN. For
rigidity, we need only recall that very large convex embeddings are rigid, [CM]. m
Corollary 2.2.10. Let B be a commutative semiprime fring with identity. Let A
be a rigid convex fsubring of B. Let M E Maxi(A) be such that M = N n A for
some N E Maxt(B). Then M is ramified in A if and only if N is ramified in B.
Let A* denote the fsubring of bounded elements of the commutative semiprime
fring A with identity. Note that A* is convex and rigid in A. In his dissertation
[Wo], Woodward proves the following fact:
Theorem 2.2.11. Let A be a semiprime fring with identity and bounded inver
sion. Let M be a maximal ideal of A and let M be the unique value of A* containing
Mn A*. The map M I+ gives a homeomorphism between Max(A) and Max(A*).
That is, Max(A*) is the subspace consisting of values of 1 in A*. In particular, if M
is a real maximal ideal of A, then M n A* E Max(A*).
Corollary 2.2.12. Let A be a commutative semiprime fring with identity satisfy
ing the bounded inversion property. Let M E Max(A) be real. Then M is ramified
if and only if I = M n A* is ramified in A*.
2.3 Rank
The first character on Spec+ (G) that we consider is simply one which counts
the minimal prime subgroups contained in a convex fsubgroup.
Definition 2.3.1. The rank, rkG(H) of a convex Isubgroup H < G is the cardi
nality of the set of minimal prime subgroups of G contained in H. If that cardinal
is not finite, then we will say that H has infinite rank; we may choose to specify the
cardinal when its value is of significance in a discussion. If H is a minimal prime
subgroup of G, then we define rkG(H) = 0.
This is a variation of the following definition given in [HLMW]: Let A be a
commutative fring with identity and M a maximal Gideal of A. The rank of M,
denoted rkA(M), is the cardinality of the subspace of minimal prime ideals of A
contained in M. By convention, if the rank of M is infinite and we don't necessarily
care about the exact cardinality, we write rkA(M) = oo. The rank of a point p E X,
rkx(p), is the rank of Mp. The rank of the fring A is the supremum of the ranks
of the maximal eideals of A, when it exists; the rank of a space X is the rank of
C(X).
We begin with illustrations of the extremal values of ranks. An egroup is
semiprojectable if for any g, h E G+, (g A h) = G(gL U b). In [BKW, 7.5.1], it
is proved that G is semiprojectable if and only if each prime subgroup contains
a unique minimal prime subgroup, which is equivalent to rkG(P) < 1 for every
P E Spec+(G). Thus, it is evident that a space X is an Fspace if and only if C(X)
is semiprojectable which is equivalent to rk(C(X)) < 1. In particular, X is a Pspace
if and only if C(X) is von Neumann regular, which is equivalent to rk(C(X)) = 0.
The onepoint compactification of the natural numbers, aN, is an example of a space
for which C(X) has infinite rank, [GJ, 14G]. In fact, if a is the point at infinity,
then the maximal ideal corresponding to a contains 2C minimal prime ideals one
for each free ultrafilter on N. Moreover, by [HJ, 4.8], this subspace of minimal prime
ideals is homeomorphic to the corona, /N \ N. Proposition 2.3.3 describes a general
situation in which we have infinite rank.
We recall some definitions. From [LZ, 39.1]: let G be a vector lattice, v E G+,
and let {g,}=1 C G be a sequence. We say that the sequence converges relatively
uniformly to g E G along the regulator v, and write gn 14 g, if for every e > 0 there
exists Ne > 0 such that for all n > N,, we have that Ig gnl ev. The sequence is
relatively uniformly Cauchy with respect to v if for every e > 0 there exists Ne > 0
such that for all n,m > Ne, we have that Ig, gn < ev. G is called uniformly
complete if for every v E G+, every sequence which is relatively uniformly Cauchy
with respect to v relatively uniformly converges along the regulator v.
Lemma 2.3.2. Let G be a uniformly complete vector lattice with weak order unit
u E G+. For any set {gj}jl, there exists g E G such that g = nfEW gf.
Proof: Let g = E', 2(lgjl A u). a
Proposition 2.3.3. Let G be a uniformly complete complemented vector lattice with
weak order unit u E G+. If for some Q E Spec+(G) we have rko(Q) > w, then Q
contains at least 2C minimal prime subgroups.
Proof: Note that, by [CM, 2.2], Min+(G) is compact since G is complemented. Let
P = {Pn)nEN be a countably infinite set of minimal prime subgroups which are
contained in Q which is discrete in the hullkernel topology on Min+(G). We first
show that P is C*embedded in Min+(G) and conclude that the minimal prime space
contains an homeomorphic copy of #N. Then we describe the elements of Min+(G)
that correspond to the points in this copy of #N \ N.
Let A, B C P be completely separated in P and index them by I, J C N
as A = {A : i E I} and B = {Bj : j J}. Fix i E I and let A, E A. For each
Bj E B, Let ai, E A+ \ Bj and bj, E Bj \ Ai. Then Ai E njJ U(bj) V' Ki and
B C UjEJ U(oj) = U(EjEJ 2i(4 A u)) V Li. Then Ki, Li are disjoint closed sets
in Min+(G).
For each i E I, generate the disjoint pair Ki, Li. Then A C cl(USIK,) V1 K
and B C niElLi = l L. By Urysohn's Lemma, the disjoint closed sets K,L are
completely separated in Min+(G) since Min+(G) is normal. Consequently, A, B are
completely separated in Min+(G) and therefore P is C*embedded in Min+(G) by
the Urysohn Extension Theorem. Finally, by [GJ, 6.5], the closed subset of Min+(G)
of minimal prime subgroups in Q contains an homeomorphic copy of #N.
Let U be a free ultrafilter on N. For g E G, let N(g) = {n : g E P,}. Define
a new prime subgroup P by g E P if and only if N(g) E U. We show that P is a
minimal prime subgroup. The following proof is the same as that for [HJ, 4.8] and
for [HLMW, 4.1].
Let g, h E P. P is a subgroup since N(g h) D N(g) n N(h) E U implies
g h E P by filter properties. By convexity and since the P, are prime subgroups,
N(gVh) D N(g) E U and N(gAh) D N(g) E U, we have that P is a sublattice of G.
Thus P is an Isubgroup of G. Let 0 < g < h E P. Then N(g) D N(h) E U since each
P, is convex, and thus P is convex. Let gAh E P, then N(g) UN(h) N(gAh) E U
implies that N(g) or N(h) is in U since U. Thus P is a prime subgroup of G.
Let g E P. Since P, is a minimal prime subgroup for each n, we have that for
each n E N(g), there exists an hn E G \ Pn such that g A hn = 0. By Lemma 2.3.2,
there exists h E G such that h = nEN(,, ) hn. Then g A h = 0 and h Pn for all
n N(g) since h' C hI C Pn for each n E N(g). Thus N(g)n H(g) V U and hence
h V P and P is a minimal prime subgroup of G. a
Recall that a space X is cozerocomplemented if for any cozeroset U C X
there exists a cozeroset V C X such that Un V = 0 and U U V is dense in X.
A concrete example of a maximal ideal of infinite rank is found in C(X) where
X is cozerocomplemented and first countable. DeMarco shows in [Le, 2] that
rk(Mp) 2 2, for any nonisolated point of a first countable space (the result actually
says more than this, and we will discuss this in Chapter 4). By modifying DeMarco's
proof, we show that Mp contains infinitely many minimal prime ideals and hence
has rank at least 2'.
Proposition 2.3.4. Let X be first countable and let p E X be nonisolated. For
every m E N there exists a family of m distinct prime ideals which sum to M,.
Moreover, if X is also cozerocomplemented, then there exist at least 2C minimal
prime ideals contained in M,.
Proof: Since p has a countable base, there exists g E C(X)+ such that Z(g) = {p}.
Let {Vi,}i be a neighborhood base at p and define a sequence of real numbers {a,}
recursively as follows: for each i, let ai E g(Vi) such that 0 < < a3 < a2 < al
and limn+oo an = 0. Let {xn} C X be a sequence of distinct preimages under g such
that xi E Vi. Then the sequence {xn} converges to p and may be considered as a
discrete set in X.
Let m > 1 be given and let Ui,U2,... ,Um be distinct free ultrafilters on the
sequence {xn}. Define Pi = {f E C(X) : 3A E Ui,A C Z(f)}, for each i. DeMarco
shows that each of these sets is a prime zideal of C(X) and that My is the sum
of any two. Thus these prime ideals are noncomparable. We will demonstrate, as
DeMarco has done for m = 2, that Mp is the sum of these m noncomparable prime
ideals.
Let {Ai,}1 be a collection of m pairwise disjoint subsets of {xn} such that
Ai e U, for each 1 < i < m. If B, = g(Ai), then Bi U {0} is a closed subset of R
Thus B, U {0} is a zeroset of K Choose Wi E C(R) for each 1 < i < m such that
Z(p,) = B, U {0} and E', i = a.
Let ui = pig. Then A, c Z(g) = Z(ui), hence u, E Pi. Finally, we have
g = ul + u2 + . + Urn. If h E Mp, then Z(h + g) = {p} and Z(h + g) = {p} By
the above, h+ + g, h + g E i Pi and hence h+, h, h E !, P.
The final statement of the proposition follow from the previous proposition,
since the cardinality of /N is 2C, by [GJ, 9.2]. *
Now that we have illustrated the extreme cases of 0,1 and infinite rank, we
present a result of [HLMW, 3.1], which gives a test for finite rank of a point of a
compact space.
Proposition 2.3.5. Let X be a compact space. Then p E X has rank n < oo if
and only if there exist n pairwise disjoint cozerosets {Uj} 1 with p E njl cl(Uj),
and no larger family of pairwise disjoint cozerosets has this property.
An fring A is called an SVring if A/P is a valuation domain (i.e. principal
ideals are totallyordered) for every prime ideal P. A space X is an SVspace if
C(X) is an SVring. We will discuss this class of rings in more detail in Chapter 3.
However, using the above, it is shown in [HLMW, 4.1] that any compact SVspace
has finite rank. The validity of the converse of this result is unknown. Presently,
our objective is to show that the result in Proposition 2.3.5 does not hold for infinite
rank. To demonstrate this, we define a cardinal function on compact spaces and
compare its value with a known cardinal invariant.
Definition 2.3.6. Let K be a cardinal and let X be a compact space with p E X. Let
{U,,}a< be a family of pairwise disjoint cozerosets of X and call the set fl<, cl(UA)
a Kboundary. Define p(p, X) to be the infimum over all (infinite) cardinals n such
that p is not contained in a Kboundary and let p(X) be the supremum over all the
points p E X of the cardinals p(p, X).
Recall that the cellularity of a space X, denoted c(X) is the infimum over
all (infinite) cardinals K such that every family of pairwise disjoint open sets of X
contains at most rK many sets.
Proposition 2.3.7. Let X be compact. Then p(X) < c(X)+.
Proof: If p(X) > c(X)+, then there exists a sboundary of cardinality greater
than the cellularity of the space, which is nonsense. m
Example 2.3.8. Let 7 > Ro. The product space, 21, where 2 is the twopoint
discrete space is called the Cantor space of weight 7. We show in Example 4.3.4
that every point of 2' has infinite rank, thus rk(C(2')) > 2' by Proposition 2.3.3.
By [E, 3.12.12(a)], we have that c+(2T) = Ri. Thus, Proposition 2.3.7 gives us that
Ro < p(2T) < R1. Therefore, since Ri $ 2c, we see that p(2T) $ rk(C(27)). O
2.4 Rank via Z#Irreducible Surjections
We must first recall a few definitions from [Ha] and [HVW2]. Let X, Y be
Tychonoff Hausdorff spaces. Let f : Y + X be a subjective continuous map. Then
f is perfect if it is a closed map such that the inverse image of any point is compact.
A perfect map is irreducible if proper closed sets of Y map to proper closed sets
of X. The pair (Y, f) is a cover of X if f is a perfect irreducible surjection from
Y to X. Let (Y1,fl) and (Y2,f2) be covers of X. We define (Y1, f) ~ (Y2,f2) if
there exists a homeomorphism g : Yi + Y2 such that f2g = fi. Order the set of
,equivalence classes of covers via: (Y1, fi) ( (Y2, f2) if and only if there exists a
continuous map g : Y1 + Y2 such that f2g = fi. A class of spaces C is a covering
class if for any space X there exists a least cover (Y, f) of X such that Y E C.
The minimal extremally disconnected, basically disconnected and quasiF covers of
compact spaces are described in [PW], [V], and, respectively, in [DHH], [HudP], and
[HVW1].
Certain covering maps allow us to compute the rank of a space externally.
A perfect irreducible surjection 4 : Y + X is Z#irreducible if for each cozeroset
U C Y, there is a cozeroset V C X such that cly(U) = cly(q'(V)). This condition
on maps is also known as sequential irreducibility and wlirreducibility. It turns out
that a map 4) is Z#irreducible if and only if C(4) is a rigid embedding of C(X) inside
C(Y), by [HaM, 2.2]. Hence we have a homeomorphism Min(C(Y)) ^ Min(C(Y))
via contraction, by [CM, 2.3]. It is therefore not surprising that these maps are
useful for calculating rank.
Example 2.4.1. Let X be a compact space. The quasiF cover of X, (QFX, Ox),
constructed in [HVW1] has the property that 4x is Z#irreducible. We summarize
this construction.
Let Z#(X) = {clxintx(Z) : Z E Z(X)}. For A E Z#(X) denote the set
of ultrafilters on Z#(X) containing A by A. The authors of [HVW1] show that
T(Z#(X)), the compact space of ultrafilters on Z#(X) whose topology has a closed
base given by {A: A E Z#(X)}, is quasiF and define a perfect irreducible surjection
x : T(Z#(X))  X by a E T(Z#(X)) maps to the unique point in n{A: A E a}.
Then QFX = T(Z#(X)) with the map bx is the quasiF cover of X.
The map bx is Z#irreducible: It is shown in [HVW1, 2.9] that if we have
A e Z#(QFX), then qbx(A) E Z#(X), which is equivalent to the property of Z#
irreducibility. In fact, the quasiF cover of X is characterized up to equivalence
in [HVW1, 2.13] as the only cover (Y, f) of X for which Y is quasiF and f is
Z#irreducible. 0
Before we continue, we discuss the question (now answered) which was our
motivation for considering this line of investigation. Let X be a compact space of
finite rank and W = (z E X : rkx(x) > 1}. In [La2], Suzanne Larson asks if W is
always closed in X. The answer is no. Her counterexample, presented at ORD98 (a
conference on 1groups held in Gainesville, FL in 1998) follows:
Example 2.4.2. Let U be a free ultrafilter on N. Let E = N U {a} where points
of N are isolated and neighborhoods of a are of the form U U {a}, where U E U.
Let Ej = E for j = 1,2 and define Y = (E1 II E2)/(al ~ a2). Let Yr = Y for each
r E R, and let X = (UJe Yr) U {oo} where neighborhoods of oo contain all but
countably many copies of the Yr. Then oo is a Ppoint which is in the closure of the
set W= {x E X: rkx(x) > 1}. D
We provide a characterization of points of finite rank of a compact space X
via Z#irreducible maps onto X. Let B be an fsubring of A and let 0 denote the
natural surjection Max(A) + Max(B). The following is proved in [HaM, 2.5].
Lemma 2.4.3. Let A and B be commutative frings with identity and bounded
inversion. Let B be an fsubring of A. Then if B is rigid in A, we have that
O(OM) = n{O(N) n B : ON = OM}, and if ON, = OM (for j=1,2) with N II N2,
then (O(N) n B) 1 (O(N2) n B).
Let X and Y be compact spaces and q : Y + X a Z#irreducible map. Then
S: Max(C(Y))  Max(C(X)) is given by M i+ {f E C(X): fob E M}.
Lemma 2.4.4. Let X and Y be compact and 0 : Y + X a Z#irreducible map.
Let p E X then O(Mq) = M, if and only if O(q) = p. Therefore, we have that
Op = n{o, n C(X) : q E 1p}}.
Proof: Let q E O'{p}. If f E O(Mq) then foq E Mq and therefore we have that
0 = fof(q) = f(p) and f E M,. Let g E Mp; then (go)(q) = g(p) = 0, and hence
g4 E O(M,). Thus O(Mq) = Mp. Conversely, assume O(q) = r 9 p. By complete
regularity, there exists f E C(X) such that f(p) = 0 and f(r) = 1. We have
f E Mp, but foq(q) = f(r) = 1 0 0 and hence f O0(M,). The final statement then
follows from the first and Lemma 2.4.3. *
Proposition 2.4.5. Let X and Y be compact spaces and f : Y + X Z#irreducible.
1. Ifp E X such that rkx(p) = n, then t1{p}) = n.
2. IfY is an Fspace and l({p}l = n then rkx(p) = n.
3. If Y has finite rank and {'{p} = n, then rkx(p) < oo. Explicitly, we have
that rkx(p) = E=i rky(qi) where 4l{p} = {fqi}l.
4. If p E X is an Fpoint of X, then q E 4 {p} is an Fpoint of Y.
Proof: (1) Let '1{p} = {qi}iEI for some index set I. For each i E I, choose
Qj E Min(C(Y)) such that 0, C_ Qi. Then by Lemma 2.4.4, it follows that we
have Op = ni~(O, n C(X)) C Qi n C(X) C Mp, for every i E I. By the bijection
described in Proposition 2.2.7, the set of minimal prime ideals contained in M, is
given by {Qi n C(X)}SEI and III = n.
(2) Let l1{p) = (qi}l=. If Y is an Fspace, then Oq, is prime for each i and we have
that Op = n!=x(0, n C(X)) C Oq, n C(X) C M for each i. Thus, by the bijection
described in Proposition 2.2.7, the subspace of minimal prime ideals contained in
Mp is {Oq, n C(X)}L1 and rkx(p) = n.
(3) Let 4r{p} = {q}?=1 and let the subspace of minimal prime ideals in Mq be
given by {Qi, : 1
ideals contained in Mp is given by {Qi n C(X) : 1 < j < rky(qi)}~1 and hence we
see that rkx(p) = Eni1 rky(qi) < oo, as desired.
(4) Let p E X be an Fpoint and 4b{p} = {q}. Then O, = O, n C(X) C Mp is a
minimal prime. There exists a unique Q E Min(C(Y)) such that Op = Q n C(X).
Since Oq n C(X) = Q n C(X) and Q is unique, Oq = Q E Min(C(Y)). U
If Y = QFX in Proposition 2.4.5, then the third statement is a partial
converse of [HLMW, 5.1] which states that if X is compact and has finite rank then
QFX has finite rank. The final statement is an extension of [HVW1, 3.12] in which
it is shown that the preimage of a Ppoint of X is a Ppoint of QFX. The second
statement says that if QFX is an Fspace, then the points of X of rank one are
precisely the points with unique preimage under the covering map Ox. In this light,
one should ask when a quasiF cover of a space is an Fspace.
Recall that a space X is fraction dense if the classical ring of quotients of
C(X) is rigid in the maximal (Utumi) ring of quotients of C(X). In [HVW1, 2.16],
it is demonstrated that QFX is basically disconnected if and only if X is cozero
complemented. In fact, the basically disconnected cover is the quasiF cover in this
case, see [HaM, 2.6.2]. The fact that QFX is realized by the extremally disconnected
cover if and only if X is fraction dense is proved in [HaM, 2.4]. By [HudP, 6.2], QFX
is an Fspace if and only if for any two disjoint cozerosets C1, C2 C X, there exist
Z, Z2 E Z(X) such that Ci C Zi for i=1,2 and int(Z n Z2) = 0.
We now provide an example of a space X such that QFX is an Fspace which
is not basically disconnected. Recall that a space is acompact if it is a countable
union of compact spaces.
Lemma 2.4.6. Let X be a noncompact acompact locally compact Fspace which
is not basically disconnected. Let X1 = X2 = #X and define Y be the quotient
space of the topological sum of X1 and X2 where pairs of corresponding points of
Xj \ X (j=1,2) are collapsed to a single point. Then Y is not quasiF and QFY is
an Fspace which is not basically disconnected.
Proof: The disjoint union U = X II X is a dense cozero set of Y which is not
C*embedded in Y. Thus Y is not quasiF. The quasiF cover of Y is X1 II X2,
which is an Fspace but not basically disconnected. *
Example 2.4.7. Let X be the disjoint union of a countable number of copies of the
corona [IN \ N. Then X is a acompact noncompact Fspace which is not basically
disconnected. Construct Y as defined in Lemma 2.4.6. Then QFY is an Fspace
which is not basically disconnected. O
2.5 Prime Character
The second character we consider counts the minimum number of minimal
prime subgroups that we must sum in order to obtain the lex kernel of an Ggroup.
Definition 2.5.1. Let S be a family of minimal prime subgroups of G. We call S
ample if VS = lex(G). The prime character of G, denoted Ir(G), is the least cardinal
so that any family of distinct minimal prime subgroups of that cardinality is ample.
Note that iflex(G) = 0, i.e., if G is a totallyordered group, then we say 7r(G) = 1.
A prime subgroup properly contained in lex(G) is called embedded.
Proposition 2.5.2. Let G be an igroup and m a positive integer. The following
are equivalent:
1. r(G) = m < oo.
2. m = 1 + sup{rk(P) : P E Spec+(G) is embedded}.
3. m is minimal with respect to the property that lex(G) = G(UjI aj) for any
m pairwise disjoint positive elements, {a}T, C_ lex(G)+.
4. m is minimal such that for any embedded prime P, the chains of proper polars
in P have length at most m 1.
Proof: (1) =* (2) : Let P be embedded. If P contains the m minimal prime
subgroups {Qj}(g C Min+(G) then V Qij C P C lex(G). Hence r(G) > m. Thus
we have shown that 7r(G) < m implies rk(P) < m. Thus by (1), rk(P) < mn 1 and
sup{rk(P) : P embedded} < m 1. If sup{rk(P) : P embedded} < m 1, then for
any family S of m 1 minimal prime subgroups of G, VS is not embedded since
rk(VS) > m1. Thus VS = lex(G) and 7r(G) 5 m 1. Thus 7r(G) = m implies that
sup{rk(P) :P embedded} > m 1. Therefore sup{rk(P) : P embedded} = m + 1.
(2) =* (3): Let {ai,}im C lex(G)+ be pairwise disjoint. Let g E lex(G) \ G(UJ afj)
and let V be a value of g such that G(Ujt= aJ) C V C lex(G). Then by the polar
characterization of O(V), [BKW, 3.4.12], we have that aj 0 O(V) for each j such
that 1 < j < m. By (2), V contains at most m 1 minimal prime subgroups,
{Qi,}Y1. Since O(V) = (_1' Qi, we have by the pigeonhole principle that there
must be a minimal prime subgroup, Q, contained in V which fails to contain two of
the elements of {aj}=1. But since these elements are pairwise disjoint, this contra
dicts the fact that Q is prime. Therefore, lex(G) = G(U,_J a%).
For the minimality of m, let P be an embedded prime of rank n < m.
Then by the Finite Basis Theorem, P/O(P) contains n pairwise disjoint elements
of corresponding to elements of P which are not in O(P), say {bk}i=1 C P. Then
bC C P for each k such that 1 < k < n and G(U = b6~) C P C lex(G).
(3) =: (1) : Assume that i(G) > m. Then there exists S = (Qj}im= C Min+(G)
which contains m elements and is not ample. Let Q = VS C lex(G). For each
j, let q, E Qf \ Qj. Then qi = Vjlqiy E Q \ (Uo# Qk). Disjointify by defining
qi = Aiojqj Alfqk V Qi. Then W E Qi for every j 9 i and then we obtain that
G(U i, i') c VS C lex(G). Hence, Ir(G) < m. By the minimality of m in (3) and
by (1) = (3), we must have 7(G) = m.
(3) 4 (4) : Follows directly from the Finite Basis Theorem [D, 46.12] applied to
P/O(P) for any embedded prime P. *
The following is immediate:
Corollary 2.5.3. Let G be an lgroup. The following are equivalent:
1. 7(G) < oo.
2. sup{rk(P): P embedded} < oo.
3. There exists m E N such that lex(G) = G(Uj'Li af) for all families of m
pairwise disjoint elements {aj)}T1.
We now consider some grouptheoretic properties of the prime character.
Note that for Igroups A and B, AEBB denotes the Igroup Ax B with componentwise
operations and is called the cardinal sum.
Proposition 2.5.4. Let G be an group.
1. For any ehomomorphic image, H, we have r(H) < Ir(G).
2. For any C E (G), ir(C) < r(G). If r(G) < oo, then r(C) = r(G) if and only
if C contains every minimal prime subgroup of G.
3. If r(A),7r(B) < oo00 and G = A B, then r(G) < r(A) + r(B) 1.
Proof: (1) Let Vp : G  H be an esurjection with kernel K. Then by [D, 9.11],
the prime subgroups of H correspond to prime subgroups of G containing K. Thus,
by the characterization of prime character in terms of ranks of prime subgroups, we
have that ir(H) < 7(G).
(2) This result follows from [BKW, 2.4.7] and [D, 27.8].
(3) Let G = AffB and let P be a prime subgroup of G. Then P = (PnA)OE(PnB),
by [D, 27.8]. Hence P contains at most (m 1) + (n 1) = m + n 2 minimal
prime subgroups and therefore r (G) < m + n 1, as desired. m
Recall from [D, 36.1] that a class C of groups is a radical class if G E C
implies the following:
1. ((G) C C,
2. every 1isomorphic image of G is in C, and
3. if {AA}AA C ~ (G) n C, then VAEAA E C.
In view of Proposition 2.5.4, it is natural to ask if the class of all Igroups of finite
prime character is a radical class. The answer is no; the following is a counterexam
ple.
Example 2.5.5. We construct an Igroup with two convex Isubgroups, A and B,
each of finite prime character such that the supremum A V B has infinite prime
character.
FA ; c  3\ IFBB
S(2.1)
Let F be the root system above, where the subgraphs FA and rF each have infinitely
many identical branches descending from its maximal vertex. Define G = E(F, R),
A = {v E G : supp(v) C rA} and B = {v G : supp(v) C Fa}. Then we have that
A G E(FA, R), B E(rB, R) and A V B E(FA U rB, R). Now, it is evident that
ir(G) = oo, ir(A) = 3 = 7r(B), and 7r(A V B) = oo. O
Recall that an tgroup G has a finite basis if it contains a finite maximal set
of elements {bj}'=1 such that the set {g E G+ : g < bj} is totally ordered for each
j. The following indicates when we can expect 7r(A V B) < oo:
Proposition 2.5.6. Let G be an fgroup and let A, B be convex esubgroups such
that ir(A) = m < 00 and 7r(B) = n < 00. If lex(A V B) = lex(A) V lex(B), then
r(A V B) < oo. Otherwise, 7r(A V B) < o00 if and only if each of A and B has a
finite basis.
Proof: Assume that lex(AVB) = lex(A)Vlex(B) and let P C lex(AVB) be a prime
subgroup of G. Then either lex(A) Z P or lex(B) Z P, or both. Say lex(A) Z P.
Then P n lex(A) is an embedded prime subgroup of A, and since 7r(A) = m, we
have that rkA(P n lex(A)) < m 1. Then by [BKW, 2.4.7], rkAVB(P) < m 1.
Likewise, if lex(B) Z P then rkAvB(P) < n 1. Thus, for every prime subgroup
P C lex(A V B), we have that rkAvB(P) < max{m 1, n 1} < oo. Therefore, by
Proposition 2.5.2, 7r(A V B) < 00.
Note that we always have that lex(A) V lex(B) C lex(A V B). We assume
now that lex(A V B) : lex(A) V lex(B) and let P E Spec+(A V B) have the property
that lex(A) V lex(B) C P C lex(A V B). If each of A and B has a finite basis, then
rkAVB(P) = IMin+(A)I + IMin+(B)I < oo and for any embedded prime subgroup Q
of AVB, we have that rkAvB(Q) <: IMin+(A) + Min+(B) Thus, by Corollary 2.5.3,
we have that ?r(AVB) < oo. Conversely, 7r(AVB) < oo implies that rkAVB(P) < 00.
Hence, since IMin+(A)I, IMin+(B) < rkAvB(P), we have that each of A and B has
a finite basis by the Finite Basis Theorem. U
The proof of the following is evident:
Proposition 2.5.7. Let G be an group and let A, B E Q(G). If A C B = lex(B)
or if lex(A) = A and lex(B) = B, then lex(A V B) = lex(A) V lex(B).
We now compare the property of finite prime character to Conrad's Property
F: every element g E G+ exceeds at most a finite number of disjoint elements. The
following is compiled in Conrad's Tulane Notes, [C]:
Proposition 2.5.8. Let G be an tgroup. The following are equivalent:
1. G has Property F.
2. Every bounded disjoint set in G is finite.
3. For every element g E G, the convex isubgroup G(g) has a finite basis.
4. Every element of G is contained in all but a finite number of minimal prime
subgroups.
Corollary 2.5.9. Let F be any root system in which each maximal element lies
above a finite number of minimal elements. Then E(F, R) has Property F.
Proof: Let v E (F, R). Let C be a maximal chain in F and let the associated
minimal prime subgroup be He = {v E E(r,R) : 7 E C =* v(') = 0}. If v c He,
then there exists 7 E C such that v(7) / 0. Since supp(v) is fnite, v is in all
but finitely many minimal prime subgroups of E(r, R). Thus by Proposition 2.5.8,
E(P, R) has Property F. *
Proposition 2.5.10. If G is a finitevalued Igroup of finite prime character, then
lex(G) has Property F.
Proof: Let g E lex(G). Then any minimal prime subgroup of lex(G) not containing
g is contained in a value of g. Since each value of g contained in lex(G) contains a
finite number of minimal prime subgroups and there are only finitely many values
of g, there are only finitely many minimal prime subgroups of lex(G) not containing
g. Thus lex(G) has Property F. a
Proposition 2.5.11. Let G be an igroup and let m be a positive integer. If m
is minimal such that every pairwise disjoint subset of G contains at most m 1
elements, then 7r(G) = m.
Proof: Any proper prime subgroup of G is contained in a value of G, hence proper
prime subgroups of G contain at most m 1 minimal prime subgroups. Thus
7r(G) < m. Let n < m and assume that every family of n minimal prime subgroups
of G is ample. Then every proper prime subgroup (in particular, every value) of G
contains at most n 1 minimal prime subgroups. This contradicts the minimality
of m. Thus 7r(G) = m. m
Example 2.5.12. The following is an example of an Igroup of finite prime char
acter such that lex(G) has Property F but G has pairwise disjoint sets of any size
m. Thus the converse of Proposition 2.5.11 does not hold.
Let F be the root system:
AAA (2.2)
and let G = E(r, R). Then ir(G) = 3 and G is finitevalued. Thus, by Proposi
tion 2.5.10, lex(G) has Property F. We demonstrate that there are bounded disjoint
families of any given size.
Index the maximal elements of F by pj, where j is a positive integer, and let
vj E G be such that vi(pj) = 1 and supp(vi) C {1y E F : 7 < pj} where vi(6) = 0 for
all 6 < pj. Then {vj}j>l is an infinite pairwise disjoint family in G. Let a positive
integer m be given. Choose any m elements from this set, {vji, 2j,,..., vi,} and let
S = Ul supp(vj,). Let v be the characteristic function on the finite set S, then
v E (F,R) and v > vj, for k = 1,2,...m. O
Example 2.5.13. The converse of Proposition 2.5.10 does not hold. That is, we
present an example of a finitevalued group with Property F and infinite prime
character. Consider the following root system F which is indexed by the positive
integers:
A A (2.3)
Then each prime subgroup of the fgroup G = E(F, R) contains a finite number of
minimal prime subgroups, yet there is no bound on the number of minimal prime
subgroups in each prime subgroup. Thus 7r(G) = oo. G has Property F by Corollary
2.5.9. 0
2.6 Filet Character
The third character that we define measures the length of a chain of incidents
of branching.
Definition 2.6.1. Let G be an fgroup. C = {P,Q,j E Spec+(G) : i > 0,j > 1} is
called a filet chain of prime subgroups if Po D P1 D Pa..., for all i, Pi 1 Qi, and
Pi+F V Qi+l C Pi for all i > 0 (see below).
P,
Q
Q (2.4)
The length of a filet chain is given by l(C) = max{j E N : 3Qj E C}. If the
maximum does not exist, we write 1(C) = oo. The filet character O(G) is given by
O(G) = sup{l(C) : C is a filet chain}. If Spec+(G) has no filet chains, i.e., if G is
semiprojectable, then we say that O(G) = 0.
Proposition 2.6.2. Let G be an egroup. Then O(G) < 1 if and only if 7(G) < 2.
Proof: Suppose that 7r(G) < 2. If O(G) > 1 then there exists a filet chain C of
length 2 in which we may assume that Po = lex(G). Since rk(Pi) = 2, we have
that 7r(G) > 2, by Proposition 2.5.2. Conversely, assume that 7(G) > 2. Then
there exist minimal prime subgroups P2, Q2 such that P1 = Pa V Q2 9 lex(G). Thus
for any minimal prime subgroup Qi Z P1, the set C = {Po, P,P2,Q1, Q2}, where
Po = lex(G), is a filet chain of prime subgroups in G of length 2. Therefore O(G) > 1.
For larger filet character, the relationship between it and the prime charac
teris more complicated, as the following example illustrates.
Example 2.6.3. Let I be the following root system:
(2.5)
Then G = E(r, R) has =(G) = 2 while 7r(G) = 4. O
The following relationships between the characters hold:
Proposition 2.6.4. Let G be an Igroup. Then
1. Ob(G) < rk(G) 1.
I. O(G) < 7r(G) 1 < rk(G).
Proof: Let rk(G) = m. If G has a filet chain C = {Pi, Qj E Spec+(G) : i > 0,j >
1}, then rk(Po) < m and hence 1(C) < m 1. Therefore, f(G) < m 1 = rk(G) 1.
If 7r(G) = n and C = {Pi, Qj Spec+(G) : i > 0,j > 1} is a filet chain
in G then 1(C) < oo since rk(Pi) 5 n 1 by Proposition 2.5.2. Hence, in fact,
1(C) < n 1 and therefore, O(G) < n 1 = 7r(G) 1. Now, 7r(G) < rk(G) + 1 by
Proposition 2.5.2. Thus, finally, 4(G) _< r(G) 1 < rk(G). n
At this time, any statement that we make about the filet character of an
group requires a restriction on the rank and prime character. Rather, we can
not say much more than what we establish earlier in this chapter. We leave this
investigation for a later date.
CHAPTER 3
GENERALIZED SEMIGROUP RINGS
Let A be a root system. Starting from the standard semigroup ring and Hahn group
constructions and the investigation of the following section, we build an fring R
having A as the skeleton of the graph of Specq(R). We follow up on ideas presented
in two papers of Conrad [C1],[C2], in the paper of Conrad and Dauns [CD], and
in the paper of Conrad and McCarthy [CMc]. The first two papers look at the
conditions on A which will yield a ring structure on V(A, R) and on the subgroup
F(A, R) of elements v for which supp(v) is the join of finitely many inversely well
ordered sets in A. The paper [CD] focuses on the case when V(A, R) is a division
ring, while in [CMc] the conditions are established for the ring to be an Iring its
properties are studied when A is finite. Note that F(A, R) is denoted by W(A, R)
in [C1],[C2], and [CD].
3.1 Specially Multiplicative fRings
Let r be a partially ordered group which is a root system. Suppose also that
F is torsion free, that the subgroup H of F generated by the positive cone is totally
ordered and that F/H is finite. In the paper of Conrad and Dauns [CD, 2.2], it is
shown that V(F, R) = F(P, R) and that V(T, R) is a latticeordered division ring
under the usual groupring multiplication: for u, v E V(r, R), and y E F
u* (7)= E u(a)v(3).
a+0=7
It is easy to see that if an element of (V(F,R),+, ) is special (i.e., has only one
maximal component), then its multiplicative inverse is also special. Hence, the
special elements of the ring V(I, R) form a multiplicative group. Moreover, the
following holds in general, [CD, Theorem I]:
Theorem 3.1.1. Let R be a latticeordered division ring with identity. The follow
ing are equivalent:
1. The special elements of R form a multiplicative group or the empty set.
2. If a E R+ is special, then a1 > 0.
3. If a E R is special, then a1 is special.
4. For all a E R+ special and x, y E R, a(x V y) = ax V ay.
Since the authors of the paper [CD] seek an embedding theorem for ffields,
their investigation in this realm is restricted to division rings. In this section, we
consider a class of frings (which are not division rings) in which the special values
form a partially ordered semigroup. The pursuit of a characterization similar to
Theorem 3.1.1 of Irings satisfying this condition is left for another time. For the f
rings that concern us here, some particular assumptions are needed on the associated
semigroups. As a formality, we define:
Definition 3.1.2. Call an fring A specially multiplicative if the special values of
A, with an appropriately adjoined 0, form a partially ordered semigroup.
Recall that an fring A is called an SVring ifA/P is a valuation domain (i.e.,
the set of principal ideals is totally ordered) for every prime ideal P. Let 9 be the
class of commutative frings which are local, bounded (that is, A* = A), semiprime,
finitevalued, finite rank and square root closed SVrings with identity and bounded
inversion. We demonstrate that the elements of G are specially multiplicative and
then investigate the properties of the associated semigroups. We must first remind
the reader of a couple of facts about special values and of the relatively deep theorem,
recorded as [HLMW, 2.14]:
Theorem 3.1.3. Let A be an fring of finite rank with identity and bounded inver
sion which is local, semiprime and square root closed. Then A is an SVring if and
only if whenever 0 < a < b and b is special, there exists x E A such that a = xb.
The proof of the following lemma is wellknown and routine, however, it is
instructive, so we include it here. We remind the reader that we denote the root
system of values of an tgroup G by F(G) = {V, : y E F}, where F is a partially
ordered index set that is orderisomorphic to F(G).
Lemma 3.1.4. Let G be an tgroup and let a, b E G+ be distinct special elements
with values at Va, Vp, respectively.
1. Va i VO if and only if a A b = 0.
2. Va < Vf if and only if a < b.
Proof: First, assume that Va II Vp. We show that a A b is contained in each value
of G and conclude that a and b are disjoint. Let V be a value in G and assume
that b V V. Then V C V# and a E V since, else, V C Va, which contradicts the
assumption of incomparability. Thus, by convexity we obtain that a A b E V. Again
by convexity, if b E V then a A b E V. Thus a A b = 0, as desired. Conversely,
assume that a and b are disjoint and, by way of contradiction, assume (without loss
of generality) that Va < Vs. Since a V Va and Va is a prime subgroup, b E Va < V,
a contradiction.
Second, assume that Va < Vs. Then a E Vs. If there exists an integer n such
that na > b, then since b V Vs, we must have that na V Vs, by convexity. But this
contradicts that a E Vs, and so we conclude that a < b. Conversely, assume that
a < b. We know that b V Va and hence Va < Vp. If Va = Vs, then by [D, 12.6], there
exists an integer n such that na > b, which is nonsense. Thus Va < VS. m
Combining Theorem 3.1.3 and Lemma 3.1.4 leads to the main result of this
section:
Theorem 3.1.5. Let A E g.
1. If a, b E A+ are special and not disjoint, then ab is special.
2. If a, a' are special with value Va, b, are special with value V, and Va, V, are
comparable, then the special value of ab is the same as that of a'V.
Proof: Let a, b E A+ be special with values at Va, Vp, respectively, and let M
be the maximal ideal in A. Since A is bounded, we may assume without loss of
generality that Va, Vp < M. This gives rise to the relations a, 6 < 1 and ab < a, by
Lemma 3.1.4(2).
Theorem 10.15 of [AF] states that an abelian Igroup is finitevalued if and
only if each positive element is a finite disjoint sum of positive special elements.
Thus we assume, by way of contradiction, that ab = cl + c2 + + c, where each
ci E A+ is special and ci A c, = 0 for all i # j. Then 0 < c < ab < a for each i
and therefore Theorem 3.1.3 gives the existence of xi E A+ such that ci = arx for
each i = 1,..., k. Without loss of generality, we may assume that 0 < zi < b, by
replacement with xz A b. Assume that az = Eii xi is a decomposition of xi into
a sum of special elements. Then ci = Ej axij implies that axi, = 0 for all but
one j, by [AF, 10.15]. Thus we may choose each az to be special, without loss of
generality.
For each i, let V&i be the value of zi. Then b V Vi, and therefore for all i, we
have that V6Y < V#. If V,% < VO, then xz < b. This gives a contradiction, since it
implies that axz < ab. On the other hand, if V, = Vp, then there exist n, m E N
such that xi < nb and b < mai. Therefore, axi < nab and ab < maxi. Thus, by [D,
12.6] we have that the value of ci is the unique value of ab and ab = ar,. Hence, ab
is special.
For the second statement, let Vh be the value of the special element ab. We
show that V5 is the value of al and then a similar argument and transitivity gives
that Vs is also the value of a'b, as desired. Towards this end, let V, be the value of
al.
If V, 1i Vs, then 0 = ab A ab = a(b A b) and hence 0 = a A (b A b). On the
other hand, if V, < V#, then since b A Y V Va, we have that a E Va. If Vp < Va,
then since a V Vp, we have b A and hence b or V is in Vp. These contradictions
lead us to conclude that Vs < VT or V, < Vs. If Vj < V, then ab << al. However,
by [D, 12.6], there exists an integer n such that nb > V and therefore, nab > nab.
Likewise, Vy Vs. Thus V, = Vs, as desired. *
Let A E G. Then by [AF, 10.10], all the elements of r(A) are special since
A is finitevalued. Abusing notation, we now identify values with their indices and
define an operation on r. Let a, b E A be special elements with corresponding values
at a, #, respectively. Append 0, a generic symbol, such that II1 a for all a E r and
define a multiplication on r U {4} such that a = = = a * and
a = if a I3,
the value of ab otherwise.
By Theorem 3.1.5, this operation is welldefined. Some properties of this operation
are recorded in the following proposition. Define a ~ P if and only if a and # are
contained in the same maximal chains of F. Then ~ is an equivalence relation on r.
Let rf denote the equivalence class of a.
Proposition 3.1.6. Let A E G and let a, #, 6 E r correspond to the values of the
special elements a, b, c, d E A+, respectively. Then
1. The operation is associative, for every a, #,7 : (a ) = a ( 7y)
2. If M is the maximal ideal of A, then M is a value of the identity. Let p = M.
Then pt a a = a p= a for every a < p.
3. If a #0 4 then a / < a, P.
4. If a < P and 7 is comparable with a, then a 7 < 3 7 and 7 a < 7 3.
5. If a $p, then a a < a.
6. Maximal chains in r are closed under the operation.
7. For each a E F, the equivalence class rF is closed under the operation.
8. If < a, then there exists / E r such that 7 = a P.
Proof: (1) Note that the products under consideration are all 4 if any of the factors
multiplied are actually 4. If a j\ P then aAb = 0. Thus aA bc = 0 and hence a I /7
which implies that (a /) 7 = 4 7 = a = a (/3 7). Likewise, if / 7 = 4 then
both products are equal to 4.
Assume that a p 964 and /37 / 4). Then either (a /) 7 = 4, or it is the
value of the special element (ab)c = a(bc). Thus, it suffices to show that a 3 P 7 if
and only if aI / '7. If a I P3 7 then aA bc = 0 and hence 0 = acAbc = (aA b)c and
so 0 = (a A b) A c. Therefore, for any value ir of a A b, we have 77 11 7. But a 3 < rl,
so a p  7. The converse follows similarly.
(2) Since A is bounded and has bounded inversion, the maximal ideal is a value
of the identity, by [Wo, 2.3.4]. Hence, it follows that p a = a p is the value of
la = al = a, namely a.
(3) Assume, without loss of generality, that / < a < p. Since ab << b, we obtain
that a/3 < 3
(4) If a < p then a < b and thus ac < be and ca < cb. By Lemma 3.1.4(2), this
just says that a. 7 < p/ 7 and 7 a < 7 /, as desired.
(5) Since a2 < a, we know that a a < a.
(6) Assume, without loss of generality, that / < a. Let 6 < P be such that 6 I a /.
Then abAd = 0 and 0 < d < b. Thus by Theorem 3.1.3, there exists y E A such that
d = yb. Then 0 = ab A yb = (aA y)b gives that 0 = aA yAb and hence 0 = ayb = ad
and therefore aAd = 0. This says that a [ 6, a contradiction. We now have that a 
is comparable with any 6 < P. It follows from this fact and from (3) that maximal
chains are closed under the operation.
(7) Let 7, 6 E r,. By (6), we have that 7 6 is in the same maximal chains as 7 and
6. Hence 7 6 E Fa.
(8) We know that 7 < a if and only if Vy < Va if and only if 0 < c < a. This implies
that there exists a positive element b such that c = ab, by Theorem 3.1.3. We have,
by [AF, 10.15], that b is special. Thus, if Vp is the value of b, then 7 = a P, as
desired. *
Corollary 3.1.7. Let A E .. Then A is specially multiplicative. Moreover, the
operation is a surjection onto F(A) U {$}.
3.2 rSystems and LSystems
Let A be a given root system. The discussion of the previous section and
of the papers [Cl], [C2], [CD], and [CMc] prepare us to construct an fring having
A orderisomorphic to its root system of values. However, under the assumptions
placed on A in these papers, these rings sometimes are erings, and rarely are frings;
in fact, as mentioned before, the paper [CD] focuses on the case when V(A, R) is a
division ring. Also, contrary to our intentions, [CMc] concentrates on the properties
of the ring when the root system is finite. We modify all these conditions on the
root system in order to get a ring multiplication, *, yielding an fring structure
on R = (F(A, R),+, *) such that the subring R = (E(A, R), +,*), as defined in
Chapter 2, is an fsubring. We start with [C1,Theorem I] which establishes the ring
structure.
Proposition 3.2.1. Assume that A is endowed with a subjective partial binary op
eration + : A  A defined on A C A x A. Let u,v E V(A, R), and define for
6 e A, u v(6) = Ia+ u(a)v(/). Then both R and R are closed under *. In fact,
R and R are rings if and only if the operation on A is (BaerConrad) associative:
(a, 3), (a +3, 7) E A if and only if (3, 7), (a, 3 +7) E A, and if (a, /), (a+/, 7) hA
then (a + #) + 7 = a + (# + 7).
Certain properties of these rings, R and R, are completely determined by the
operation + on A; commutativity is one such attribute.
Definition 3.2.2. Let (A, +) be a root system with a partial binary operation, +.
We call (A, +) an rsystem if the operation is surjective and (BaerConrad) asso
ciative. An rsystem (A, +) is commutative if (a, #) E A if and only if (/, a) E A
and a +/ = P + a for all such pairs.
Proposition 3.2.3. Let (A, +) be an rsystem. (A, +) is commutative if and only
if 7 is a commutative ring.
Proof: If (A, +) is commutative, then for u, v E R we have
u*v(6)= u(a)v() = E v(#)u(a)= v(8)u(a)= v *u(6).
a+0=6 a+=6 5+a=
Thus 1Z is a commutative ring.
Conversely, if a + /3 /3 + a, then Xa+# = Xa X,= + X8 Xa = X#c+a.
In order to obtain an Iring, we must ask that the operation on A preserve
the order of the root system and restrict the domain of the operation. As we will
indicate, the various strengths of order preservation and restrictions of domain yield
varying richness of order structure on the rings.
Definition 3.2.4. Let A be a root system. An rsystem (A, +) is an isystem
if it satisfies: a < / and (a, 7) E A implies that (#,7) E A and, in this case,
a+,7 < /#+7; and if (7, a) E A implies that (7, #) E A and, in this case, 7+a < 7+/3.
If every connected component of A has a maximal element, we call A bounded above.
Note that the Isystem condition gives that no nonmaximal element of such
a bounded above root system is idempotent. The following theorem is in [CMc, 2].
Theorem 3.2.5. Let (A, +) be an rsystem. Then 7' is an Iring if and only if A
is an Isystem.
In the next section, we describe the conditions on an Isystem which induce
an fring structure on %. As mentioned, there are some restrictions that we need
to place on the domain and range of the operation. Before we discuss this situation,
we present an example of an Isystem that gives rise to an Iring which is not an
fring. Probably, this is the simplest example of such an Isystem.
Example 3.2.6. Let A > a > 2a > 3a... and p > 8 > 2/ > 3# .... Then
we let Aa = {A} U {na}O1, let A# = {p} U {n/p}01 and totallyorder A4, x Ap
lexicographically such that (61, 71) < (62, 72) if and only if S6 < 62 or we have 61 = 62
and 71 < 72. Identify pairs with sums and let Aa+, = {6+7 : (6, 7) E Aa x A#}. We
define an associative and commutative addition on the root system given by disjoint
union A = An II A, II Aa+ as shown in the following table, where v = A + p and
k, 1, m, n are positive integers (note that the table is completed by reflection across
the diagonal):
+ a A p v A+ms ka+p noa+I,
a 2a a+6 a a+ a+p a+m (k + )a+p (n+l)a +1
0 2 A+ 0 A+ A+(m+ 1) k+# na+(TI+1)
A ___ A v v A+m ka+p na+I1
p _V + m ka + + nao +
v V' A+mO ka+ p noa+I
A +mS A+ 2m# ka +m na + (m +)
ka+ p 2ka+p + (k + na + I
na + 10 2na + 21P
Let u, v E R+. We need to show that u v E R+ in order to conclude that R
is an Iring. Since u and v are positive, it is evident that u*v(A), u*v(p), u*v(v) > 0.
If 0 = u v(A) = u(A)v(A), then assume that u(A) = 0. For an integer n,
u v(na) = u(A)v(na) + u(na)v(A) + u(ia)v(ja).
i+jn
ij0
If u(na) = 0 for all n, then u v(na) = 0. Conversely, if n 0 0 is minimal such that
na E supp(u), then u v(na) = u(na)v(A) > 0. Thus, if v(A) > 0 then na is the
maximal element of supp(u v) below A.
If v(A) = 0 and v(ma) = 0 for all m, then u v vanishes on A,. Otherwise,
let m be minimal such that ma E supp(v). Let k = n + m then ka is maximal in
supp(u v) and u v(ka) = u(na)v(ma) > 0. A similar computation demonstrates
that u v > 0 at the maximal element of its support in A# also.
Assume that 0 = u v(v) and, when they exist let
Au = max(supp(u) n A.) Av = maz(supp(v) n A,)
= max(supp(u) nL A) p = maz(supp(v) n A#)
6u = max(supp(u) n Aa+#) 6v = maz(supp(v) n A,+O) (3.1)
Then the maximal element of supp(u v) lying below v is given by
7 = TmaX{A, + p ~,A+ 64, A, + Alu, \a + 64, pu + 4, p, + 6 + 6t}
and so we are left to show that u v(7) > 0. We have three cases to check here,
namely 7 = A + m/, ka + p or noa + I. Upon consideration of the formulae below, it
quickly becomes dear that the only nonzero summands of u v(7) are those of the
form u(a)v(r), where a and r are one of the six maximal support elements listed
above in (3.1), which implies that the summands are all positive. For the sake of a
certain degree of completeness of exposition, we list the possible summands in the
case that 7 = A + mp. The analysis in the other cases is similar.
u v(A + mp) = u(A)v(m#) + u(mn)v(A) + u(A)v(A + mp) + u(A + m/)v(A)
+ u(/A)v(A + mp) + u(A + m8p)v(() + u(v)v(A + mr) + u(A + mr)v(v)
+ E u(A+ip)v( ) + E u(i/)v(A+jp) + E u(A+i i)v(A+j j)
i+j=m i+jm i+j=m
ijo ijso ijsA
To compute u v(A + mp), consider all the possible combinations of the
following situations and then add.
1. If u(A) # 0 then if v(mn) # 0, we have that m6 < p,. If mr < ,, then
7 = A + m/ < Au + i which is nonsense. Thus m/ = pu, in this case and
u(A,)v()p,) is a summand of u v(7).
2. By an argument similar to the above, if v(A) # 0 and u(mp) # 0, then
mfn = p and u(p,)v(A,) is a summand.
3. If u(A) # 0, u(/) / 0 or u(v) # 0 then if v(A + m/7) # 0, we may conclude
that 6, = A + mp and u(Au)v(6J,), u(pu)v(,,) or, respectively, u(6,)v(6,) is a
summand.
4. Similarly, if v(A) # 0, v(/i) # 0, or v(v) # 0 then 6, = A +mpf and u(6,)v(A,),
u(6u)v(p,), or u(6,)v(6,) is a summand.
5. If, for some i + j = m, we have that u(A + iP)v(jP) # 0, then we also have
6, = A+iPif,, p = jp and u(6,)v()(t,) is the only nonzero contribution from this
large sum.
6. Likewise, if u(iP)v(A+j#/) # 0 for some i+j = m, then u(p,)v(6,) is the only
nonzero component of this summation.
7. Finally, if u(A+i)p)v(A+j/) # 0 then u(6u)v(6,) is the only nonzero summand
coming from this summation.
As stated before, the analysis in the other two cases is similar. We provide
the formulae below:
u v(ka + p) = u(ka)v(p) + u(p)v(ka) + u(A)v(ka + p) + u(ka + p )v(A)
+ u(s)v(ka + p) + u(ka + p)v(p) + u(i)v(ka + p) + u(ka + p)v(v)
+ E u(ia + p)v(ja) + E u(ia)v(ja + I) + u(ia + J)v(ja + p)
i+j=k i+j=k i+j=0
is3o idj# ij#o
u v(na + I1) = u(na)v(l1) + u(l/3)v(na) + u(A)v(na + I1) + u(na + 1ft)v(A)
+ u(pi)v(na + 10/) + u(na + l13)v(p) + u(v)v(na + 1/) + u(na + lIf)v(v)
+ u(na)v(A +1 0) + u(A + l/)v(na) + u(na + it)v(l1) + u(l/)v(na + ip)
+ u(ia + lf)v(ja) + E u(ia)v(jU + l) + u(na + iP)v(j#)
i+j=n i+j=n i+j=l
ij#O ijAo ijAo
+ u(ip/)v(na +j) + C u(ia + i)v(ja + 1) + E u(ia + 1)v(ja + p)
i+j=l i+j=n i+j=n
ijZo iJ9o idjo
+ u (A + ip)v(na + j#) + u(na + i/)v(A + ji)
i+j=L i+j=
ij#o ij4o
+ E u(ia + kpl)v(ja + mni)
i+j=n
k+m=I
ijk,mv0
R is not an fring: Let HI+ = {v E R : 6 E Aa+6 =, v(6) = 0} be the
minimal prime subgroup. Then X,, X# E HA+,, yet Xa *X# = Xa+6 V HA.+ Thus,
HA+o is not an ideal and R is not an fring by [BKW, 9.1.2]. ]
The following two propositions record some consequences of the Isystem
condition. Note that none of the excluded conditions occur in Example 3.2.6.
Proposition 3.2.7. Let (A, +) be a bounded above system in which maximal ele
ments act as an additive identity on elements below it. Then the following can not
occur for nonmaximal elements a, 3 E A,
1. 8 + a < a, where P, a are below the same maximal element, and / II P + a
2. 3
3. 53<0+a
4. If A, pI A are maximal and (A, M) e A, then A + p A1 it.
In particular, the second and third properties imply that if a, P, 3+a are comparable,
then 3+ a < a, 3.
Proof: If P + a < a,/  + a, and p > a, 3 is maximal, then X,, Xu Xa are
positive. Yet, X# (X, Xa) = X6 X#+a is negative, contradicting that R is an
Iring.
Assume that / < a < 3 + a. Let i > p + a be maximal. Then as above,
X# (Xi, Xa) < 0. The same contradiction is obtained in the case that we assume
13+ 3+a
If A, p are maximal and A + p < p then Xx (X, 2X\+,) = Xa+p < 0. U
Proposition 3.2.8. Let (A, +) be an system.
1. There is no nonmaximal 6 E A such that 6 is idempotent and 6 + a = 6 for
all nonmaximal a > 6.
2. If 6 A is nonmaximal and idempotent then there does not exist a maximal
element p > 6.
Proof: If 6 E A such that 6 is idempotent and 6+ a = 6 for all nonmaximal a > 6,
Then we contradict the assumption that R is an ring since Xj (Xa 2X6) = X6.
If p > 6 is maximal and 6 is idempotent, then X ( (X, 2Xs) = XS. U
3.3 fSystems
We are much more interested in the fring situation. In [CMc, 2.1], the
authors demonstrate a condition on A which will give rise to an fring.
Theorem 3.3.1. Let (A, +) be an tsystem. Then R is an fring if and only if the
root system also satisfies: if a  p and (a,'y) E A, then a+7 II 3; and if(7,a) E A,
then 7+a II p.
In this section we consider a condition on A which is equivalent to the one
stated above and proceed to investigate certain properties of the associated frings.
Recall that r(F(A,R)) is orderisomorphic to A by Proposition 2.1.1, where the
values are of the following form, for 6 E A :
V = {u E R: > 6 = u(y7) = 0}.
Definition 3.3.2. Let (A, +) be an esystem such that 6 = a + # implies a, f > 6.
Then we say that (A, +) is an f0system.
Proposition 3.3.3. If (A, +) is an fosystem, then % is an Iring and for every
6 E A, the subgroup V6 is an ideal. Hence, in particular, R is an fring and R is
an fsubring of R.
Proof: Assume that (A, +) is an f0system. Let 6 E A, v E Vs, u E R, and 7 6.
Let 7 = a+#f, then by the f0system assumption, # 7 6 and hence v(0) = 0, for
all such / since v E Vs. Then u* v(7) = E +=,y u(a)v(P) = 0 and we conclude that
u*v E VF. Therefore, the values of R are ideals. Moreover, since each minimal prime
subgroup is an intersection of a chain of values by [D,10.8], each is an intersection
of a chain of ideals; thus each minimal prime subgroup itself is an ideal. Finally, we
have shown that R and R are frings by [BKW, 9.1.2]. m
Proposition 3.3.4. An system (A, +) is an f0system if and only if % is an
fring.
Proof: Sufficiency is shown in Proposition 3.3.3. Conversely, assume that there
exist a, / E A such that a + /3 a or a + / /t Then by Proposition 3.2.7, we
have that a, /, a + / are not all comparable. If a I /, then we have that a I a + /
by Theorem 3.3.1. Assume that / j a + P and let C be a maximal chain in A
containing a + /. Let He = {v E % : 6 E C = v(6) = 0} be the associated minimal
prime subgroup. Then XB E He, yet Xa X, = Xa+ Hc. Thus He is not an ideal
and hence 17 is not an fring by [BKW,9.1.2]. *
Definition 3.3.5. An f0system satisfying the following is called an fsystem:
1. If a and # are comparable, then (a, /), (0, a) E A.
2. If p is maximal, then (6, p), (p, 6), (/s, If) E A and 6 + p = p + 6 for every
6 < p. In particular, p. + = p.
We will shortly see that these additional assumptions on a bounded above fo
system make maximal chains in A into monoids. This is quite useful in our setting.
For instance, it is not difficult to figure out when the frings have a multiplicative
identity, under the fsystem assumption.
Proposition 3.3.6. Let (A, +) be a bounded above fsystem. R and 1Z each have
a twosided multiplicative identity if and only if A has a finite number of connected
components.
Proof: Let (A, +) contain only a finite number of connected components with
maximal elements {i, #2, ..., p,}. Let Xj be the characteristic function on the set
{pj} and let e = E = X. Then for v E R~ and 6 E A,
v e(6) = E v(a)e() = v(6)e(k) = v(6),
a+O=6
where pk > 6 is maximal. Likewise,
e v(6) = e(a)v() = e(k)v(6) = v(6).
a+B=6
Thus e is a twosided multiplicative identity.
Conversely, assume that e E 7I is an identity and let p E A be maximal with
characteristic function X,. Then 1 = X,(p) = e xm,() = e(p). Thus, supp(e) has
as many maximal elements as there are maximal elements in A. Since e E r?, we
must conclude that there are only finitely many connected components in A. U
It is handy to have the following definition:
Definition 3.3.7. Call an rsystem (A, +) unital if the ring 7 has a multiplicative
identity.
The rings R and R are semiprime, as we will now show. Thus the minimal
prime subgroups are also prime ideals, by [BKW, 9.3.1].
Proposition 3.3.8. Let (A, +) be an f0system. Then % is semiprime.
Proof: Let u E 7, where u = EjEi ajXj for an index set I of supp(u) and for
a, E R\0 for all j E I. Then u*u(6) = 6=,+6j aiaj. Let 6i be maximal in supp(u).
We show that 6i + 5i is maximal in supp(u u). Let 6 = 6j + 8k E supp(u u).
If 6j 1I Si, then by the f0system condition, 6 1 6 + 65. On the other hand, if
Jj, 6k < 6, then 6 < Si + i, as desired. Thus u* u(6i + 6) = ai > 0 and we conclude
that 7I is semiprime. *
Corollary 3.3.9. Let (A, +) be an f0system. Then maximal chains in A are closed
under the operation +.
Proof: Let C C A be a maximal chain and let He = {v E : 6 E C = v(6) = 0}
be the associated minimal prime subgroup. Let a, ~ E C and assume by way of
contradiction that a + p f C. Then X, X# = Xa+# E Hc but Xa, X# 0 He. Thus
He is not a prime ideal. This contradicts Proposition 3.3.8, by [BKW, 9.3.1]. m
One should ask if the maximal eideals of 7 are actually the maximal ideals;
or equivalently, one should ask if R has the bounded inversion property. The answer
is yes, if (A, +) is a unital fsystem.
Proposition 3.3.10. Let (A, +) be a bounded above unital fsystem. An element
u E R is a multiplicative unit if and only if u 0 V, for all maximal p E A. Thus 1
satisfies the bounded inversion property.
Proof: Let S C A be the set of maximal elements of A and let e = EEs Xi be
the multiplicative identity. If u E R is a multiplicative unit, then there exists v E R
such that u v = e. Hence 1 = e(p) = u(L)v(/p) for every i E S. Thus u(p) 0 0 for
all 1 E S and hence u V, for all maximal p E A, as desired.
Assume that u V, for all maximal p E A. We define the multiplicative
inverse v of u recursively on each maximal chain in the support of u. Let v(6) = 0
for all 6 V supp(u) and let v(p) = 1/u(ip) for each maximal p E A. Then if 61 < / is
maximal in supp(u), we just solve the equation 0 = u*v(61) = u(61)v(pL)+u(p)v(61)
to get that v(61) = u(61)/u(p)2. Proceed with the definition of v accordingly. That
is, let p > 6 E supp(u) and assume that v(7) is defined for all 7 > 6. Then u(a)v(r)
is a summand of u v(6) only if 6 < a, 7 E supp(u). Thus v(6) is the only unknown
in and is the unique solution of the equation 0 = u v(6).
Hence, since if e is the multiplicative identity of R and u > e then u 0 V,
for all maximal p, we have that u is a multiplicative unit. Therefore 7 satisfies the
bounded inversion property. *
Let (L, <) be a partially ordered set. Recall that A C L is called closed if
{ai}ier and Aiai or Viai exists in L then Aiai, Viai E A. It is the case that, if (A, +)
is an fsystem, then all of the closed convex esubgroups of 7 are ring ideals. We
will use the following special case of [D, 45.26].
Theorem 3.3.11. Let G be a finitevalued egroup. Then there is an orderpreserving
correspondence between the closed convex esubgroups of G and (order) ideals $ of
the root system r(G) given by
K 14 (K = {G6 E F(G) : 3k E K such that Gs is a value of k}
4~ = {=g G : all values of g are in <}.
Proposition 3.3.12. Let (A, +) be a bounded above fsystem. Any closed convex
subgroup of R is an ideal of R.
Proof: Let K be a closed convex Isubgroup of R; let u e K+ and v E R. If u is
a unit, then 4)K = r(G) and by the preceding theorem, K = G. So assume that u
is a nonunit.
Let p E A be maximal and assume that u v(7) = 0 for all Jt > 7 > 6 and
u*v(6) # 0 for some 6. By the preceding theorem, we need to show that 6 E IK C A.
Let 6 = a + p such that u(a)v(3) # 0. Then there is a maximal a' > a such that
u(d) # 0. Then a' = a'+ a + > a+/ = 6 anda' E K Since u K.
Therefore, we conclude that 6 E
by the preceding theorem. *
We now seek the prime ideals and zideals among the prime subgroups Vs
and their associated value covers. The cover of V8 is the set
P = {v ER: v(7) = for all 7>6}
and is the smallest convex Isubgroup properly containing V, and Xs. First let us
recall the most general definition of zideal. Let G be a vector lattice, v e G+,
and let {g,}n=L C G be a sequence. Recall that the sequence converges relatively
uniformly to g E G along the regulator v and write gn _+ g, if for every e > 0
there exists N, > 0 such that for all n > N6, we have that Ig gI < ev. Let
H be a convex Isubgroup (subvector lattice) of G. The pseudoclosure of H is
H' = {g E G : 3{gn,},1 C H, gn + g for some v E H+}. Then H is relatively
uniformly closed if H = H'; let H denote the relative uniform closure of H. Then
if G(g) denotes the convex Isubgroup (subvector lattice) of G generated by g,
we define a convex fsubgroup (subvector lattice) H to be an abstract zideal if
h E H,g E G and G(g) = G(h) imply that g E H. In fact, [HudPI, 3.4] says that
this definition is equivalent to G(h) C H for all h E H.
Proposition 3.3.13. Let (A, +) be an fsystem. If p E A is mazimal, then Vc is
a prime ideal of R which is an abstract zideal.
Proof: Let u v E V,, and u V V,. Assume without loss of generality that
u+, + E V, and u f V,. We show that v E V,. Since u f V,, we have that
u(p) 0. Thus since u+,v+ e V,,
S= u *(p) = u(a)v() = (+ u)(a)(v+ v)(0),
a+0=0 a+#=I
and since p is maximal, we have that 0 = u(Is)v(I) and therefore, v~(p) = 0.
This gives that v E V, and v E V,. Hence V, is a prime ideal.
Since I/V, is isomorphic to R via the evaluation map u "* u(t), we have
that V, is uniformly closed by [HudPI, 2.1]. Thus by [HudPI, 3.4], V, is an abstract
zideal. n
Corollary 3.3.14. If (A, +) is a bounded above unital fsystem, then the mazimal
ideals are given by {V, : pE A is maximal}.
Proposition 3.3.15. Let (A, +) be an fsystem and let 6 E A be nonmaximal.
Define PS = {v E R : v(7) = 0 for all 7 > 6}. Then Ps is an abstract zideal and
it is a prime ideal if and only if a + P > 6 for all a, > 6.
Proof: Let v E P1. Then there exists {vn}il C Pj such that v, 4 v, for some
w E R+. Let 7 > 6, then for every e > 0 we have that lvl(7) < ew(y). Thus v(7) = 0
and Pa is relatively uniformly closed. Therefore, PS is a zideal by [HudPI, 3.4].
Assume that Pg is a prime ideal and that there exist a, / > 6 such that
a + P < 6. Then Xa+, E P6 and Xa+6 = Xa XB. But Xa, XB 1 PS, which is a
contradiction.
Conversely, assume that a+# > 6 for all nonmaximal a, # > 6. Let u*v E PS
and assume, by way of contradiction, that u, v Pg. Then u v(7y) = 0 for all
7 > 6 and there exist elements a E supp(u) and # E supp(v), each maximal in the
support set and such that a,3 > 6. Assume, without loss of generality that a > P. If
a' < a, ff 6 / and at least one of the inequalities is strict, then a' + ff < a + / and
we conclude that u*v(a+P) = u(a)v(p) +u(P)v(a). If a = /, then since a+ / > 6,
we have that 0 = u v(a + f) = 2u(a)v(/) t 0, a contradiction. If a > f, then
v(a) = 0 and hence 0 = u v(a + /) = u(a)v((/) 6 0, another contradiction. Thus
we conclude that either u or v is in P8 and therefore P6 is a prime ideal. *
Let S be a totallyordered set. Recall that S is an ri set if whenever A, B C S
are countable and A < B, then there exists c E S such that A < c < B. Since R
is not an i1set, the ring R is never an 771set. To see this, let 6 E A and consider
the sets {xj} > {(1 )x6 : n E N}. But, R can satisfy a related, slightly weaker
condition.
Definition 3.3.16. We call a totallyordered set S an almost l1set if A, B C S
are countable and ifA < B, then there exists c E S such that A < c < B. Note that
R is such a set.
Proposition 3.3.17. Let (A, +) be a totallyordered fsystem. R is an almost
r71set if and only if A is an Ir set.
Proof: Assume that R is an almost 71set. We first note that A contains no suc
cessor pair. Let a < / be a successor pair. Then the sets {nXa}neN and {1/nX#}nEN
contradict the almost riset condition.
Let A = {aI}}N, B = {lj}jEN C A, where a 2
XaY < Xa2 < ... < X& < X#i and there exists f E R such that X, < f < Xj for
all i,j. Let 7 be the maximal element of supp(f). Then f Xa, 0 implies that
7 = ai and f(7) > 1 or 7 > a. and f(7) > 0. If 7 = a,, then f X,, < 0, which
is a contradiction. Thus 7 > ai for all i. Similarly, 7 < p# for all j. Therefore, A is
an rhset.
Conversely, assume that A is an rilset and let fi < f2 < .* < g2 < gl E 1.
Let 1 < 2 < < 72 < 71 E A be the corresponding maximal elements of the
support sets. Let 4 = {}j)jEN and r = {7j}iNs. We have a few cases to consider: $
and r are the same constant sequence, one of the sequences is eventually constant,
or neither sequence is eventually constant.
If there exists n E N such that Oj = a = 7, for i,j > n, then we get the
following string of inequalities in R : fi(41) < f2('2) < * < 2(72) < 9g1(,). Since
R is an almost r7set, there is r E R such that fj(41b) < r < gi(i,) for all i,j. Then
fj < rxa < gi for all i,j. If $ is eventually constant and r is not, say Oj = a for
all i > n, for some n, then, by hypothesis, there exists P E A such that a < 0 < r.
Then fji X# < gi for all i, j. If neither sequence is eventually constant, then by the
riset hypothesis, there exists P such that 1j < f < 7y for all i and fj < XO < gi for
all i, j. M
3.4 Survaluation Ring and nfRoot Closed Conditions
Recall that a commutative ring A is a survaluation ring (or SVring) if A/P
is a valuation ring for every prime ideal P. In this section, we set down a character
ization of those fsystems which give SVrings.
Let (A, <, +, ) be a totally ordered and cancellative abelian monoid with
identity element p. We define the group of differences, qA, of A as it is done in
[Fu, X.4]. Define an equivalence relation on A x A by (i6, 62) ~ (7, 72) if and only
if 81 + 72 = 71 + 62. It is clear that the relation is reflexive and symmetric; the
transitivity follows from the cancellation property of the monoid. We let qA be the
quotient A x A/ ~, denote the class of the element (61, 62) by [61 62], and define
an operation + as is usual. That is, [J6 62] + [71 72] = [(6 + 71) (62 + 72)]. The
cancellation in the monoid ensures that the operation is welldefined. The element
[p p] is an identity and [62 61] is an additive inverse of [61 62]. We define
[61 621 5 [71 72] if and only if 61 + 72 < 71 + 62. By [Fu, X.4.4], this is the
unique order on qA which extends the order on A. Finally, oembed A in qA via
8 4 [6 p].
Let H be a partially ordered groupoid. Then h E H is called negative if
hx < x or xh < x, or both for all x E H. The groupoid H is called negatively
ordered if every element is negative.
Definition 3.4.1. Let A be a partially ordered semigroup. A is called inversely
naturally ordered if it is negatively ordered and 6 < a implies that there exists
Se A such that 6 = a + 3.
Example 3.4.2. Let A = {1 }11, U [1,oo) C R be inversely ordered with the
usual addition in the reals. Then A is an fsystem which is not inversely naturally
ordered. To see this, note that (1 1) + c = 1 if and only if c = n = 2 or
c= 1,n= 1. [
Let (A, +, p) be a totallyordered abelian cancellative monoid with maximal
element p, such that (A,+) is an fsystem. Let X = {X6 : 6 E A}. Then (X, *, X,)
is a totallyordered abelian monoid which is Iisomorphic to (A, +, p). Since X is
written multiplicatively, the elements of the group qX are quotients and we denote
them as such in the following proof.
Theorem 3.4.3. Let (A, +, p) be a totallyordered abelian cancellative monoid with
maximal element p. Let X = {xs : 6 E A} and R = F(A,R). The following are
equivalent:
1. R is a valuation ring.
2. R is 1convex.
3. I is Bizout.
4. 1 is convex in qR.
5. X is convex in qX.
6. A is convex in qA.
7. A is inversely naturally ordered.
Proof: The equivalence of (1), (2), and (3) is [MW, Theorem 1]; the equivalence
of (2) and (4) is [ChDi2, Lemma 2].
(2) = (5) : Let X, < Xa./X < Xa. Then 0 5 Xa < Xs+p and by (2), there exists
f E R such that Xa = f X+f. Iff = EC j fj3Xo, where J is some index set,
fj E R and q dj E A for all j E J, then
Xa(e)= f X+b(e)={f ife= +6+
Thus, for some j E J, we have that a = 3 + 6 + j, fj = 1 and fA = 0 for all
k 5 j. Hence f = x~ and Xa/Xp = Xs+
(5) (6) : Follows since A L X.
(5) => (7) : Let a < p E A. Then a +/ < a and therefore Xa+, < Xa < X# which
implies that Xa < XaI/X < X,. Thus, by (5), we have that Xa/Xy# = X6 E X for
some 6 E A. Therefore, Xa = X8+6 and a = P + 6.
(7) = (2) : Let 0 < u < v E 7, and assume that u = YjE~aijX,, and that
v = EkeK bkxpk for index sets J, K, and where aj, bE E R for all j E J, k E K.
Also assume that a, < 61 are maximal elements in the respective support sets.
Then by (7), for every j E J and every k E K, there exists 6i, 7y E A such that
aO = A + b and P3 = #I + 7. Note that 71 = p. Then u = XA, E* jJ aixa and
v = XA EkEK b4Xk
Let w = ZEK bhX7k. Then w(p) = bl > 0 and hence w is invertible in
R. Let z = Eij ajXs,. Then by cancellation in A, we have that 0 < x < w and
0 x< w. If we let f = (w)* (Ix), then x = w *f and = w f.
Finally, u = X, x = X, w f = v f, as desired. a
The following lemma is wellknown and routine to verify.
Lemma 3.4.4. Let (A, +) be an lsystem and C C A, a maximal chain. Denote the
associated minimal prime subgroup by He = {u E 7 : C n supp(u) = 0}. The map
cp : F(A, R) + F(C, R) given by restriction is a subjective Iring homomorphism
with kernel He. Thus, %/Hc F(C, R.)
Corollary 3.4.5. Let (A, +) be a unital fsystem. Then 7 is an SVring if and
only if each maximal chain in A is inversely naturally ordered.
Example 3.4.6. 1. If A1 = [0, oo) C R is inversely ordered with the usual
addition of real numbers, then F(A1, R) is an SVring.
2. If A2 = {1 1}=1 U [1, oo) C R is inversely ordered with the usual addition
in the reals, then F(A2, R) is not an SVring.
3. Let A = R[[x, y]] be the ring of formal power series in the indeterminates z, y.
Order the monomials lexicographically via 1 > x, y and x'yi < zxyl if and
only if k < i or k = i and I < j. The ring A is not an SVring since it is
not 1convex: note that 0 < y < x and the equation y = xf has no solution
fEA.
Let Z,, = Z = {n E Z : n > 0} be inversely ordered. In the lexorder
described above, if A = Z, x Zy, then A c F(A, R). We convexify A in qA by
convexifying A in qA. That is, if Ac = A U {(n, m) E Z x Z : n > 0, m < 0}
then F(Ac, R) ^ A({xayi : i > 0 or i = 0 and j > 0}) UA is an SVring which
is the convexification of A in qA. O
Recall that an fring A is nthroot closed if for every a E A+ there exists
b E A such that a = b". This property arises in R if there is a certain amount of
divisibility in the arithmetic structure of A.
Definition 3.4.7. Let (A, +) be an rsystem. A is called ndivisible if for every
6 E A there exists a E A such that na = 6. We say that the system is divisible if it
is ndivisible for all n E N.
Proposition 3.4.8. Let (A, +) be a totally ordered fsystem such that R is nthroot
closed. Then A is ndivisible.
Proof: Let 6 E A. Then xs = v" for some v = E ij ajXa. If 6 = p is maximal
in A, then 6 = nip, so we assume that 6 # p. If ac E supp(v) is maximal in the
support set, then nal E supp(v") is maximal. Therefore 6 = nal, as desired. w
Proposition 3.4.9. Let (A, +) be a totally ordered inversely naturally ordered f
system with maximal element p. If A is ndivisible, then R is ntroot closed.
Proof: We begin with squareroots. Let u E R+ be given by u = EjEj ajX", for
some index set J, aj E R, and j E A for all j E J.
If u(p) 6 0, then we define a squareroot v recursively on A. To begin, let
v(p) = i) and assume that a, is maximal in supp(u) \ {p}. Let v(6) = 0 for
all al < 8 < p. Then we define v(al) = u(al)/2v(p,). Let S be the Nlinear span
of supp(u) and define v(6) = 0 for all 6 V S. If v(7) is defined for all 7 > 6, then
we define v(6) to be the unique solution of the equation u(6) = ;+r=~ v(a)v(r),
where, necessarily, a, r E {a E S : a > 6}. Then u = v v, by construction.
Now, assume that u(/p) = 0 and let al be the maximal element in the support
of u. Since A is inversely naturally ordered, for every j E J, there exists ij E A such
that aj = al +6i. Note that 61 = p. Then u = Xa1 *EjE, aXj. Let w = jEJJ aXj ,,
then w(/p) # 0. Thus w = vi vl by the above construction. Since A is 2divisible,
al = 271 for some 71 E A. Therefore Xai = X, Xn. Letting v = X, vl, we then
have that v v = Xyi vl Xy, v = Xai w = u. Thus nthroots exist when n is a
power of 2.
Let n be odd and let v E R be given by v = 'keK bkX#, for some index
set K, bk E R, and flP E A for all k E K. As with square roots we consider two
cases. First assume that v(p) # 0 and define an naroot recursively. Let w(p) be a
real ntroot of v(/i) and let v(6) = 0 for all A < 6 < /, where a is the maximal
element of supp(v) \ {p}. Then the nfold convolution product equation we must
solve reduces to v(,8) = w"(A) = n(w(p))nlw(1). In order to see this, we proceed
by induction on n. If n = 2, then w*w(/31) = w(=/)+w)+w(#)w(/) = 2w(p)w(A).
Assume that the statement holds for n = m. Then
wm'(Pi) = (w w')(1) = w(u)w'm(I) + w(#i)w'm()
= w(p)m(w(/p))m1w(fi) + w(fi)(w(/A))m = (m + 1)(w(p))mw(/O)
as desired. We may then define w(/1i) = v(j1)/(n(w(p))n1).
In general, we let w(7) = 0 if 7 is not in the Nlinear span of the support of v.
Assume that w(7) is defined for all 7 > J. We show that the equation v(6) = wn(6) is
linear in w(6); hence, we may define w(6) to be the unique solution of this equation.
If n =2, then w2(6) = ,+r=, w(a)w(r) + 2w(5)w(p). Assume that wm(6) is linear
in w(6). Then
wm+1(6) = w (a)wm(7.) + w(w),m'(6) + w(6)w'(t)
or+Tr=
is linear in w(6), by induction, since w(6) will not appear in wm(r), as 6 < 7 and A
is an fsystem. Thus, in this case, v has an ntroot.
Second, assume that v(it) = 0 and proceed as in the squareroot case. Let 61
be the maximal element in the support of v. Since A is inversely naturally ordered,
for every k E K, there exists 6k e A such that /3k = f/ + 6k. Note that 6~ = p.. Then
v = X EYeK b xk Let x = EkEKbkx6,, then x(p) 0. Thus x = w:, for some
wl, by the above construction. Since A is ndivisible, #I = nai for some 71 E A.
Therefore Xp, = (X,,) Let w = X, wi; then wn = (X, Wl)" = XI z = v. a
Corollary 3.4.10. Let (A,+) be a totally ordered inversely naturally ordered f
system with maximal element p. A is divisible if and only if R is nhroot closed for
all n.
Example 3.4.11. 1. Let Ai = [0, oo) C R be inversely ordered with the usual
addition of the real numbers. Then F(A1, R) is ntroot closed for all n.
2. Let A2 = {n E Z : n > 0} be inversely ordered. Then A2 is not 2divisible
and Xi > 0 has no squareroot. That is, if Xi = v2, then v(0) = 0 and we then
conclude that 1 = Xi(1) = 2v(0)v(1) = 0, a contradiction.
3. Let A3 {1 } U [1, oo) C R be inversely ordered with the usual addition
in the reals, then A3 is not inversely naturally ordered and similarly, X1 has
no squareroot since 1 has no nonzero summand. ]
Proposition 3.4.12. Let (A, +) be a totally ordered fsystem with maximal ele
ment p. Then if p has an immediate predecessor ir, then X, has no squareroot. If
there exists p > 6 E A such that 6 has no nonmaximal summand other than itself,
then Xa has no squareroot.
Based on the preceding examples and results on ntroots, we formulate the
following:
Conjecture 3.4.13. Let (A, +) be a totally ordered fsystem with maximal element
p. F(A,R) is squareroot closed if and only if A is 2divisible and every nonmaximal
element of A has a nonmaximal summand other than itself.
Recall that a field K is realclosed if every positive element is a square and
every polynomial p E K[z] of odd degree has a root in K. An integral domain
R is called realclosed if qR is a realclosed field. Let (A, +) be a totally ordered
inversely naturally ordered fsystem with maximal element p. Assume that A is
also 2divisible. Then R is a 1convex and squareroot closed fdomain. By [ChDil,
Theorem 1], under these conditions, R is realclosed if and only if every odd degree
polynomial over R has a root. What additional assumptions on A are necessary to
guarantee the realclosed property?
Conjecture 3.4.14. Let (A, +) be a totally ordered commutative fsystem with
maximal element p. If A is divisible and inversely naturally ordered, then R is real
closed.
We end this section by shedding a little light on this conjecture. Recall from
[HLM] that a commutative fring A with 1 satisfies the Intermediate Value Theorem
for polynomials (or is an IVTring, for short), if for every p(t) E A[t], and pair of
distinct elements u, v E A such that p(u) > 0 and p(v) < 0, there exists w E A such
that p(w) = 0 and uA v < w < u V v. We show that a totally ordered commutative
semiprime valuation fdomain with identity is realclosed if and only if it is an IVT
ring. It is necessary to record the following unpublished theorem of Suzanne Larson,
which was communicated via electronic mail on April 17, 1997. Her proof follows.
Theorem 3.4.15. Let A be a commutative semiprime IVTring with identity. If
S is a multiplicatively closed subset of regular elements of A+, then the ring of
quotients, S1A is an IVTring.
Proof: Let p(t) E S'A[t] be given by p(t) = aoboW + albi't + + anbnt" and
assume that p(ulv') > 0 and p(u2v) 5< 0. Let d = v'vb2bobi .. b. Then d E S is
regular. Define a new polynomial q(t) E A[t] by
aovvb ** "bn+alvn'vbob ... bnt+a2vn2v2n2boblb3 .... bt2+.. +abob, ".
Then q(ulv2) = dp(uiv ') > 0 and q(u2vi) = dp(u2v1) < 0. Since A is an IVTring,
there exists w E A such that ulv2 A U2V1 < w < u1v2 V u2v1 and q(w) = 0. Then
u1vp1 A u2v21 < wv'v1 < u1v1 V uO21 and dp(wv'v1') = q(w) = 0. Since d is
regular, p(wv lv2') = 0 and we conclude that the quotient ring is an IVTring. m
Proposition 3.4.16. Let A be a totally ordered commutative semiprime valuation
fdomain with identity. Then A is realclosed if and only if it is an IVTring.
Proof: If A is realclosed, then qA is a realclosed field and, by [ChDi2], qA is an
IVTfield. Let p(t) E A[t] be such that p(u) > 0 and p(v) < 0, for some u, v E A.
Then there exists w E qA such that p(w) = 0 and u A v < w < u V v. Since A
is a valuation domain, A is convex in qA by [ChDi2, Lemma 2]. Hence, w E A
and we have that A is an IVTring. Conversely, if A is an IVTring, then qA is an
IVTfield, by the preceding theorem of Larson. Then, by [ChDi2], qA is realclosed
and therefore A is realclosed. *
CHAPTER 4
RAMIFIED PRIME IDEALS
In this chapter we expand on the notion of a ramified prime ideal, as defined in [Le],
which we introduced in Chapter 2. We first examine the concept in general and then
move to try to understand ramified maximal ideals which correspond to nonisolated
Gapoints. This result is then used to consider local versus global ramification
conditions.
4.1 Ramified Points
Definition 4.1.1. Let X be a completely regular space. A prime ideal of C(X) is
ramified if it is the sum of the minimal prime ideals that it contains. We define
p E X to be ramified if Mp is ramified. A point p E X is totally ramified if every
prime zideal contained in Mp is ramified. The space X is (totally) ramified if every
nonisolated point of X is (totally) ramified.
A ramified 1ideal of C(X) is a prime ideal, by Corollary 2.2.6. LeDonne
proves that a ramified prime ideal of C(X) is necessarily a zideal. Let us consider
two extreme conditions. Recall that we say a point p E X is an Fpoint if Op is
prime. Ifp is an Fpoint, then since Op is the unique minimal prime ideal contained
in Mp, Mp is not ramified. Likewise, in this case, no prime zideal contained in Mp is
ramified. On the other hand, the condition of total ramification ensures branching
at every prime zideal. Analytically, LeDonne shows [Le, 3]:
Theorem 4.1.2. If X is a metric space then every maximal ideal of C(X) is totally
ramified.
Note that this result says that every maximal ideal of C(X) (fixed or free) is
ramified, if X is metric. We do not know of any weaker topological condition which
guarantees total ramification of C(X).
Definition 4.1.3. Let A be a commutative fring with identity. For any integer
n > 2, call a prime tideal P nlimbed if P is the sum of n noncomparable prime
eideals of A which are properly contained in P. A point p of X is nlimbed if Mp is
nlimbed. Note that any nlimbed ideal P is necessarily ramified and rk(P) > n.
Example 4.1.4. We now present an example of an fring 1 in which a maximal
ideal is ramified but not nlimbed for any n. Let A0 = [0, oo) C R and define
A' = (1/n, oo) R, for i = 1,2, where each interval is inversely ordered. Let
An = A II A: be the disjoint union and then let A = Ao H (II,,,A). We obtain a
root system with the induced ordering which we describe in the following diagram:
(4.1)
We endow A with a partial commutative associative binary operation. Let
(*)i denote the sum in parentheses as the usual sum of real numbers residing in A~;
the mark "" signifies that the sum is undefined. Note, to conserve space, the table
is completed by reflection across the diagonal.
+ rEAo rEA A rE6 I rE A rE
E o (r + s)o (r + a) (r + a)' (r + _s)__ (_+ _
sE A (r + a) (r + s). (r + s) if k < n;else (r + a)if I < n; else 
E A(r +s (r + 8) if k
sEA, ________ (r + a)2
Let R = F(A,R) and V, = Ao II (IIm>nA); let Co = Ao II (IIne1A') and
C, = Dn) II A. Then the minimal prime ideals of R correspond to these maximal
chains and are given by Q, = {u E R : u(Cn) = {0}}, where n = 0,1,2,....
Any supremum VjE Qj over a finite set J C w is the prime ideal P, given by
{u E R : u(2)) = {0}}, where m is the maximum element of J. Hence, for all
n E w, the maximal ideal Vo is not nlimbed since it is not a finite supremum of
minimal prime ideals. However, Vo = Ve,, Qn, and so it is ramified. O
We show in Proposition 2.3.4 that for any nonisolated point p of a first
countable space, the maximal ideal Mp is nlimbed for every n. If the space is also
cozerocomplemented, then rk(M,) > 2C and Mp is 2Climbed. From this, we also
obtain the following, which is weaker than Theorem 4.1.2:
Corollary 4.1.5. Every metric space is ramified.
The following theorem means that if X is a metric space that is not pseudo
compact, then there exist points of /X \ X such that MP is minimal. Hence, not
every maximal ideal of C(X) branches nontrivially in the root system Spec(C(X)).
Recall that we call a topological space X perfect if every closed set of X is a Gsset.
Note that any metric space is perfect. A point of fX is remote if it is not in the
#Xclosure of any nowhere dense subset of X. A point p E #X \ X is a Cpoint
if p inttx\x(cldxZ(f) n 8X \ X) for all f E C(X). A theorem similar to the
following appears in [W, 4.40]. All the proofs there carry through here, verbatim,
under our reduced hypotheses.
Theorem 4.1.6. Let X be a completely regular space and consider the following
conditions on a point p E #X \ X:
1. p is a Cpoint.
2. Z[MP] contains no nowhere dense set.
3. MP=OP.
4. p is a remote point.
Without additional assumptions, (3) =4 (1). Let X be perfect and assume the exis
tence of a remote point, then (4) = (2) =* (3). If X is perfect and the set of isolated
points of X has compact closure in X, then (2) => (4). If X is realcompact and
Cpoints exist, then (1) =o (2).
It is not known if a remote point p always has the property that MP = OP.
We do know the following about the rank of a remote point:
Proposition 4.1.7. Let X be a completely regular space. Let p E fX be a remote
point. Then rkc(x)(MP) = 1.
Proof: In [vD, 5.2], it is demonstrated that no remote point is in the closure of two
disjoint open sets of #X. Thus, in particular, no remote point is in the closure of two
disjoint cozerosets of #X. By [HLMW, 3.1], we have 1 = rkax(p) = rkc(ox)(M*P).
Finally, since C(fJX) is rigid in C(X), we have rkc(x)(MP) = 1. U
Finally, we ask: does ramification of a point in X indicate ramification in
#X, or vice versa?
Proposition 4.1.8. A point p E X is ramified in X if and only if it is ramified in
38X. Likewise, a point p E vX is ramified in vX if and only if it is ramified in 8X.
Proof: This is a corollary of Proposition 2.2.12, since we know C(#X) = C*(X)
and MfX = Mx n C(#X). a
4.2 Ramified Gspoints
The main theorem, Theorem 4.2.5, of this section provides a good method
for checking the ramification of Gapoints. We will use it to characterize ramified
Gapoints in normal countably tight spaces and to find some ramified points in
product spaces. We first discuss the following proposition, which we will obtain as
a corollary to Theorem 4.2.5.
Proposition 4.2.1. Let p E X be a G6point. If X \ p is not C*embedded in X,
then rk(p) > 2.
Since an Fpoint has rank 1, the preceding proposition, proved in [Le] and
(in greater generality) by van Douwen in [vD], shows that a Gapoint, p, is not
an Fpoint if it has the property that X \ p is not C*embedded in X. We give a
counterexample for the converse if the G6condition is lifted.
Example 4.2.2. Let X = B2(N, /N \ N) be as defined in Example 5.2.5. There,
we show that there exists a point p of the corona such that rk(p) = 2, X \ p is
C*embedded in X and p is not ramified. But no point of the corona is a G6. [
The following two results are Theorems 2.1 and 2.2 of [K2]; we will use these
to prove our theorem on the ramification of Gapoints.
Theorem 4.2.3. Let p be a nonisolated Gapoint of X. If Z e Z[C(X \ p)] then
clx(Z) E Z[C(X)].
Define 7 : Z[C(X \ p)]  Z[C(X)] by 7(Z) = cx(Z). Let 4 be the
extension of the identity map X \ p  X to the largest subspace X1 C #(X \ p)
such that it is extendible as a continuous map into X.
Theorem 4.2.4. Let p be a nonisolated Gapoint of X then
1. The mapping 7 is onetoone from the set of prime zfilters on X\p converging
to points of 0'({p}) onto the set of prime zfiters on X contained properly
in Z[Mp].
2. A prime zfilter W on X \p converging to a point of 1~'({p}) is a zultrafilter
if and only if 7(W) is maximal in the class of prime zfilters on X contained
properly in Z[Mp].
Theorem 4.2.5. Let p be a nonisolated Gjpoint of X. The point p is ramified if
and only if X \p is not C*embedded in X.
Proof: Let p be a nonisolated Gjpoint of X. If p is not ramified then the prime
zideal P = oQEMin(M) Q is properly contained in Mp. We will show, in this case,
that every point of X is the limit of a unique zultrafilter on X \ p. Then by [GJ,
6.4], X \p is C*embedded in X.
Let q E X \ p. Then Mq E Max(C(X \ p)) gives rise to the zultrafilter
14 = Z[Mq] on X \p. Clearly q E n{clx(U) : U E t4}. The uniqueness of U4 is a
standard result [GJ, 3.18].
By [GJ, 6.3(b)], there exists a zultrafilter U on X\p converging to p. Assume
that there exists another such zultrafilter, V. Let U = yU and V = yV. Then
Qu = ZxU and Qv = Z V are prime zideals of C(X) which are properly contained
in M,. If Qu C P then U = Zx[Qu] C Zx[P]. Hence U = 'rU c rZx[P], which
contradicts that U is a zultrafilter on X \p. Likewise, Qv is not properly contained
in P. Thus P C Qu, P C Qv and by [GJ, 14.8(a)], we must have either Qu C Qv
or Qv C Qu. But Qu C Qv gives that U C V and therefore U C V. Since U is an
ultrafilter, U = V, as desired. In a similar manner, if Qv C Qu, then V = U.
Conversely, assume that p is ramified. Then Mp = E Min(Mp) and there
exists more than one prime zideal in C(X) which is maximal in the class of prime
zideals properly contained in M,. These give distinct prime zfilters on X which
are maximal in the class of prime zfilters on X properly contained in Z[Mp]. Hence,
via 7, we have distinct ultrafilters on X \ p converging to p. Again by [GJ, 6.4],
X \p is not C*embedded in X. *
Corollary 4.2.6. If X is a metric space, then X is ramified.
Corollary 4.2.7. If X is first countable and p E X is nonisolated, then X \ p is
not C*embedded in X.
Corollary 4.2.8. Ifp E X is a Gjpoint and X \p is not C*embedded in X, then
rk(p) > 2.
Let X and Y be completely regular spaces which are not Pspaces and let
W = X x Y. We conjecture that every nonisolated point of W is ramified. We use
Theorem 4.2.5 to deduce two partial answers to this question.
Proposition 4.2.9. Let X and Y be completely regular spaces and let W = X x Y.
Let x E X and y E Y be nonisolated Gapoints and let p = (x, y) E W. Then
W \ {p} is not C*embedded in W.
Proof: Since X is completely regular and x is a Gjpoint of X, {x} is a zeroset
of X. Say, {x} = Zx(f) for some f E C(X). Then we have for E, = {(} x Y,
E, = {z} x Y = 7r(Zx(f)) = Zw(f o 7x)
where lrx denotes the natural projection from W onto X. Likewise, we have that
Ey = X x {y} = Zw(g o ry) where g E C(Y) such that Z,(g) = {y} and ry is the
natural projection from W onto Y. Let and g denote the restrictions of fo7rx and
golry to W\{p}. Then we have that E,\{p} = Zw\{p}(f) and E,\{p} = Zw\ p(g)
are disjoint zerosets of W\ {p}. Thus E \ {p} and E\ {p} are completely separated
in W\{p}. But p E clw(E \{p})nclw(E\ {p}) and therefore E,\{p} and E,\{p}
are not completely separated in W. By the Urysohn Extension Theorem, W \ {p}
is not C*embedded in W. *
Corollary 4.2.10. Let X and Y be completely regular spaces and let W = X x Y.
Let x E X and y E Y be nonisolated Gapoints and let p = (x, y) E W. Then p is
ramified in W.
Proof: Follows from Theorem 4.2.5. U
Proposition 4.2.11. Let X and Y be completely regular spaces and let W = X x Y.
Let p = (x, y) E W be nonisolated. If W \ {p} is normal, then W \ {p} is not C*
embedded in W.
Proof: Let E, = {x} xY and Ey = X x {y}. Then E,\{p} and E,\{p} are disjoint
closed sets in the normal space W \ {p}. Thus, they are completely separated in
W \ {p}. But pE clw(E, \ {p}) n clw(E, \ {p}), so E, \ {p} and E, \ {p} are not
completely separated in W. Therefore, by the Urysohn Extension Theorem, W\ {p}
is not C*embedded in W. m
Corollary 4.2.12. Let X and Y be completely regular spaces and let W = X x Y.
Let p = (x, y) E W be a nonisolated Gapoint of W If W \ {p} is normal, then p is
ramified in W
Proof: Follows from Theorem 4.2.5.
We now investigate the ramification of Gspoints in a class of spaces more
general than metric or first countable spaces.
A topological space X is countably tight if for a subset U c X we have that
any p E cl(U) is in the closure of a countable set S c U. A Frechet Urysohn space is
one in which every p E cl(U) is the limit of a sequence of distinct points {pn} C U.
It is evident that any FrchetUrysohn space is countably tight.
Lemma 4.2.13. Let X be a normal topological space and let p e X be nonisolated.
Then X \ {p} is C*embedded in X if and only ifp 4 clx(A) ncldx(B), whenever A
and B are completely separated in X \ {p}.
Proof: By the Urysohn Extension Theorem, X \ {p} is C*embedded in X if and
only if A and B are completely separated in X, whenever A and B are completely
separated in X \ {p}. Assume that X \ {p} is C*embedded in X and let A and B
be completely separated in X \ {p}. Then A and B are completely separated in X
and hence are contained in disjoint closed sets of X. Thus p dclx(A) clx(B).
Conversely, let A and B be completely separated in X \ {p}. We wish to show
that A and B are completely separated in X. By hypothesis, p clx(A)n clx(B).
Thus, clx(A) and clx(B) are disjoint closed sets of the normal space X. Hence, A
and B are completely separated in X. *
Proposition 4.2.14. If X is a normal countably tight topological space and p E X
is nonisolated, then X \ {p} is C*embedded in X if and only if for every two
countable sets St and S2 which are completely separated in X \ {p}, we have that
p clx(S) n ldx(S2).
Proof: Assume that X \ {p} is C*embedded in X and let S, and S2 be two count
ably infinite sets which are completely separated in X\ {p}. Then by Lemma 4.2.13,
we have that p V dx(S) n clx(S2).
Conversely, let A and B be completely separated in X \ {p}. Assume that
p E clx(A) n clx(B). Then there exist countable sets S1 C A and S2 C B such that
p E clx(Si) n clx(S2). Since A and B are completely separated, so are S1 and S2.
This contradicts the hypothesis. Thus by Lemma 4.2.13, A and B are completely
separated in X and X \ {p} is C*embedded in X. *
Corollary 4.2.15. Let X be normal Frichet Urysohn space and let p E X be non
isolated. Then X \ {p} is C*embedded in X if and only if there do not exist two
sequences in X which are completely separated in X \ {p} and converge to p.
Finally, Proposition 4.2.14 and Theorem 4.2.5 imply the following.
Corollary 4.2.16. If X is a normal countably tight topological space and p E X is
a nonisolated Gspoint, then p is ramified in X if and only if for every two countable
sets Si and S2 which are completely separated in X \ {p}, p dclx(Si) dx(S2).
4.3 Ramification via CEmbedded Subspaces
Let A and B be commutative rings with identity. Assume that we have a
subjective ring homomorphism f : A B, with K = Ker(f). Recall that there is a
onetoone orderpreserving correspondence between Spec(B) and the set of prime
ideals P E Spec(A) such that K C P. Let M E Max(B) be such that M = P1 + P2
for some nonmaximal proper primes Pi, P2 E Spec(B). If N E Spec(A) corresponds
to M, then we have B/M (A/K)/(N/K) A/N. So N E Max(A). Let Q, Q2
be the prime ideals of A corresponding to P1 and P2. Then (Q1 + Q2)/K P1 + P2
via the surjective map given by a + b + f (a) + f(b) with kernel K. Thus, by the
correspondence, we have that N/K M = Pi+P2 ( (Q,+Q2)/K and N = Qi+Q2.
In fact, we have shown:
Proposition 4.3.1. Let A and B be commutative rings with identity such that there
exists a subjective ring homomorphism f : A  B with K = Ker(f). If P < B is a
prime ideal which is a sum of two distinct prime ideals then there exists Q E Spec(A)
such that Q/K K P and Q is a sum of two distinct prime ideals of A.
Ramification in a Cembedded subspace implies global ramification.
Corollary 4.3.2. Let X be a completely regular space and let Y be a Cembedded
subspace. If a point p of Y is 2limbed in Y, then p is 2limbed in X.
Proof: Since Y is Cembedded in X, we have a surjective ring homomorphism
from C(X) onto C(Y), given by restriction with kernel {f E C(X): Y C Zx(f)}.
Hence this result follows from Proposition 4.3.1. 0
Proposition 4.3.1 also gives the following, since any compact subspace of a
completely regular space is Cembedded.
Corollary 4.3.3. 1. Let X be compact, Y C X a closed subspace. If a point of
Y is 2limbed in Y then it is 2limbed in X.
2. Let X be a compact space consisting of more than one point. If every noniso
lated point in a proper zeroset of X is 2limbed, then every nonisolated point
of X is 2limbed.
Example 4.3.4. The Cantor Set is metric, hence every point is 2limbed. By [E,
3.12.12c], every point of 2T, the Cantor space of weight r, is contained in a closed
set which is homeomorphic to the Cantor Set. Thus every nonisolated point of 2T
is 2limbed. In fact, induction on Proposition 4.3.1 gives that every point of the
Cantor space is nlimbed for every n E N. []
Conversely, if a maximal ideal of A containing K is a sum of primes containing
K, then by the correspondence given above, its image is a maximal ideal which is a
sum of primes in B. That is, if N E Max(A), K C N, and N = Q1 + Q2 such that
K C Qi and K C Q2, then f(N) E Max(B) as B/f(N) G (A/K)/(N/K) G A/N.
And f(N) = f(Qi + Q2) = f(Q1) + f(Q2) Q1/K + Q2/K.
This gives a partial converse:
Proposition 4.3.5. Let A and B be commutative rings with identity such that there
exists a subjective ring homomorphism f : A + B with K = Ker(f). Let P be
a prime ideal of A containing K. Then P is a sum of two distinct prime ideals
containing K if and only if f(P)/K is a prime ideal which is a sum of two distinct
prime ideals of B.
Corollary 4.3.6. The ringhomomorphic image of a commutative ramified ring
with identity is ramified.
Corollary 4.3.7. Let Y be a Cembedded subspace of X and let p E Y have finite
rank in X. If p is ramified in Y, then p is ramified in X. If any function in C(X)
vanishing on Y also vanishes on a neighborhood of p, then p is ramified in Y if and
only if p is ramified in X.
Proof: The first statement is an application of Proposition 4.3.1 by induction. The
second statement follows from Proposition 4.3.5 by induction. a
Note that the hypothesis of the second statement of the above merely de
mands that the kernel of the restriction map be contained in Opx. This is satisfied
if Y is open or if Y is a Pset. The preceding results indicate that ramification is a
local property.
Let A be a commutative ring with identity, let S be a multiplicative system in
A such that 1 E S. Then there exists a onetoone correspondence from Spec(S1A)
onto {P E Spec(A) : P S = 0}. The proofs of the following are routine:
Proposition 4.3.8. Let A be a commutative ring with identity, let S be a multi
plicative system in A such that 1 E S.
1. If P E Spec(S'A) is a sum of two proper primes, then the preimage of P, the
set {x E A : x/1 E A}, is a prime ideal which is a sum of two proper primes
in A.
2. If Q E Spec(A) is a sum of primes and Q n S = 0, then S'Q is a sum of
primes in S1A.
87
Corollary 4.3.9. Let Y be a subspace of X such that C(X) + C(Y) is a ring of
fractions map. That is, there is a multiplicative system S C C(X) such that 1 E S
and C(Y) = S1(C(X)). Then:
1. Forp E Y, rky(p) < rkx(p).
2. Let p E Y. If Mf n S = 0, then there is P E Spec(C(X)) such that P C Mf
and rky(p) = rkx(P).
CHAPTER 5
mQUASINORMAL fRINGS
In [Lal3], Suzanne Larson defines the notion of a quasinormal fring; one in which
the sum of any two distinct minimal prime ideals is a maximal tideal or the entire
fring. We generalize this definition and a few of her results.
5.1 Definitions
Definition 5.1.1. Let A be a commutative fring with identity and let m be a
positive integer. Call A mquasinormal if the sum of any m distinct minimal prime
ideals is a maximal eideal or the entire fring A. If X is a topological space such
that C(X) is mquasinormal, then we call X an F,space.
Note that the "2quasinormal" is Larson's "quasinormal" condition, the "1
quasinormal" condition is equivalent to von Neumann regularity, and if A is m
quasinormal then A is nquasinormal for any n > m. Hence, the Fispaces are
exactly the Pspaces and any Frspace is an Fnspace, when n > m.
Theorem 5.1.5 generalizes [Lal, 3.3] and characterizes the mquasinormal
semiprime frings. Note that [Lal, 2.2], which we now state, gives necessary and
sufficient conditions for a semiprime fring to have the property that the sum of
any two distinct minimal prime ideals is a prime eideal. This condition is stronger
than the assumption we make in our theorem, but this result indicates when one
can expect to be able to apply it.
Theorem 5.1.2. Let A be a semiprime fring. The following are equivalent:
1. The sum of any two semiprime ideals is semiprime.
S. The sum of any two minimal prime ideals is prime.
3. The sum of any two prime ideals is prime.
4. For any a, b E A+, the ideal a1 + b' is semiprime.
5. For any two disjoint elements a, b E A+, the Iideal a + b is semiprime.
6. For any a, b E A+, the tideal a + b' is semiprime.
7. When x, a, b, c, d E A+, a, b 0 are such that x2 = c +d and aAc = bAd = 0,
there ezist g, h E A such that x = g + h and gAa= hA b= 0.
The theorem above holds for C(X) since the sum of two prime ideals is prime
by [GJ, 14B]. We will use the following lemmas. The first follows from the fact that
a prime fideal P of a semiprime fring is minimal if and only if for every p E P
there exists q V P such that pq = 0. The second lemma shows the existence of
certain functions which we will take for granted in the proofs to follow.
Lemma 5.1.3. Let A be a semiprime fring in which the sum of any m distinct
minimal prime ideals is a prime iideal. Let P be a prime fideal. Then P is
minimal with respect to containing J~m=' aJ if and only if for every p E P there
exists q P such that pq E =,1 al.
Lemma 5.1.4. Let A be a commutative fring. Let n > 2 and let Q, ...,Q,, be
distinct minimal prime ideals. Then there exists an element f E Qf \ Uj, Qj, for
each i = 1,...,n.
Proof: Let fk E Q+ \ Q for k = 1,..., n. Then, by convexity, we have that
f = .fi E Q+ \ Uj Qj, as desired. m
Theorem 5.1.5. Let m be a positive integer. Let A be a commutative semiprime
fring with identity in which the sum of any m distinct minimal prime ideals is a
prime Lideal. The following are equivalent:
1. A is mquasinormal.
2. For every nonmaximal prime fideal P, rk(P) < m 1.
3. Let {aj}'m1 be a family of positive pairwuise disjoint elements of A. Proper
prime Iideals containing j1= a are maximal tideals.
4. Let {aj}im1 be a family of positive pairwise disjoint elements of A, let M be
a maximal fideal containing the ideal ~1 afJ, and let p E M. Then there
exists z f M such that zp e 7iam= a.
Proof: (1) =* (2) : Let P be a nonmaximal prime ideal of A such that rk(P) > m
and let Qx,... Qm be m distinct minimal prime ideals that are contained in P. Then
mL1 Qj C P is not maximal, hence A is not mquasinormal.
(2) = (3) : Let P be a prime fideal containing Esm!_ af. Then ai C P for every
j. Therefore, aj V O(P) for all j, and hence P contains at least m minimal prime
ideals by the pigeonhole principle. Thus condition (2) gives us that P is a maximal
fideal.
(3) (4) : Follows from Lemma 5.1.3.
(3) =o (1) : Let M be a maximal fideal and let Q,..., Q C M be minimal prime
ideals. Then by hypothesis, E~j= Qj is a prime ideal and we are left to show
that it is a maximal ideal. For each j = 1,... m, let as E Q+ \ UiqQi and define
bj = AisjaiA^=a E YEj Qj. Then {bj)} = is a pairwise disjoint set of m distinct
elements of E~ 1 Qj and by the choice of the aj's, we have that Allak E nfm=lQk
by convexity and Ai,6ai V Qj for each j. Hence bj V Qj and bf 9 Qj, for each j.
Thus ESm= Qj is a maximal fideal by condition (3), since E'=l bj C jE I gj
The quasinormal condition is a variation of the normal condition, which is
that the sum of any two minimal prime ideals of a semiprime fring with identity
is the entire fring. This is discussed in [Hu]. The expected generalized definition
follows, along with a theorem recording two equivalent conditions. The result is a
special case of Theorem 5.1.5 and the proof follows immediately from the one above
and from [Hu, Theorem 8].
Definition 5.1.6. Let m > 2 be a positive integer. An fring A is mnormal if for
any pairwise disjoint family {aj}m1, we have that A = E=J= a .
Theorem 5.1.7. Let A be a commutative semiprime fring with identity and let
m > 2 be a positive integer. The following are equivalent:
1. A is mnormal.
2. For any maximal ideal M, we have that rk(M) < m 1.
3. The sum of any m distinct minimal prime ideals is A.
Before we move to describe Fmspaces, we first discuss a special class of
mquasinormal frings.
Definition 5.1.8. Let A be a local fring. An embedded prime eideal P is high
if for every minimal prime ideal N E Min(A), either N C P or N V P = lex(A).
Otherwise, P is low. Call an fring A a broom ring if for every maximal ideal M
every prime tideal in AM is high.
The following is immediate from Proposition 2.5.2 and Lemma 5.1.3.
Proposition 5.1.9. Let A be a local commutative semiprime fring with identity
and maximal ideal M. The following are equivalent:
1. A is a broom ring.
2. r(A) < 2.
3. If P lex(A) is a prime Iideal, then rkA(P) < 1.
4. If a, b E A are disjoint and P is a proper prime ideal containing a + bL,
then lex(A) C P C M.
5. If a, b E A are disjoint and a1 + bV C lex(A), then for every p E lex(A), there
exists z 4 lex(A) such that zp E a1 + b.
Proof: (1) < (2) : Since every prime fideal of A is high, we have that for any
two distinct minimal prime ideals Q, Q2 that Q, V Q2 = lex(A). Hence 7r(A) < 2
by definition. Conversely, 7r(A) < 2 implies that every minimal prime ideal is high
and therefore that every prime fideal is high, as desired.
(2) 4 (3): Since 7r(A) < 2, we have that rk(P) < 1 for any embedded prime fideal
P, by Proposition 2.5.2; and vice versa.
(2) = (4) : By Proposition 2.5.2, we have that al + b = lex(A).
(4) > (5): Follows from Lemma 5.1.3.
(4) = (3) : Assume that (4) holds. Let P be an embedded prime ideal and assume,
by way of contradiction, that rk(P) > 2. Let Qi, Q2 C P be minimal prime ideals.
Let qi E Qf \ Q2 and let q2 E Q+ \ Q1. Disjointify by defining f = qj q^ A q2
for j = 1,2. Then i,' C Q~ since ji f Q2 and q4 C Qi since V Q1. Then
'1 + 41 C P and hence lex(A) C P, which contradicts that P is embedded. *
Example 5.1.10. We now present an example of an fring which is 3quasinormal
but is not a broom ring. Let Ao = A = [0, oo) C R, and A2 = A3 = (1,oo) C R,
where each is inversely ordered. Identifying the copies of 0 in the disjoint union
A1 = (Ao II Ai)/(0o ~ 01) and letting A2 be the disjoint union A2 II A, we obtain
a root system A = A1t A2 with the induced ordering which we now describe. That
is, r < s if and only if either r, s e Aj for j = 0, 1, 2 or 3 and r < s in the inversely
ordered real numbers; or if r E A2, s E A1; or if r E A3, s E At. Explicitly, r I s if
r E A2 and s Ao I A3 or if r E At and s E Ao.
We endow A with a partial binary operation. To begin, note that we define
Oo + o0 = 01 + 01 = + 01 = Oo ~ 01. Let rj, sj E Aj be nonzero for j = 0,1,2,3.
Let (*)j denote the sum in parentheses as the usual sum of real numbers residing in
Aj; the mark "" signifies that the sum is undefined.
+ ro 8o ri a1 r2 82 rs3 s
ro (2roo o +o)o 
so (ro + ) (2o)o 
rl (2ri)i ( 1+rl)i (r2+rl)2 (a2+ r)2 (r3+r r) (3s+r)3
si (rl + i) (281) (r2 + 1) (82 + (r2 + +)s (8s+ )s3
r2 (rl +r2)2 (i +r2)2 (2r2)2 (2 + r2)2 
2 (r + 2)2 (81 +82)2 (r2 + ) (282)2 
rs (ri + ra)s (8 + rs3) (2rs)3 (83 + rs)
83 (ri + s)s (81 +3)3 (rs + 83)3 (2ss)
Let Co = Ao, Ci = Ai,C2 = A II A2, and C3 = A I A3. Then the minimal prime
ideals of F(A, R) are of the form Qj = {u E F(A, R) : supp(u) C A \ C} for
j = 0,2,3. The similarly defined Qi is a prime ideal by Proposition 3.3.15. Now,
it is evident that L = lex(F(A, R)) = Qo V Q2 V Q3 and Q2 V Q3 = Q1 # L so we
know that 7r(F(A, R)) = 3. Therefore, Proposition 5.1.9 shows that the ring is not
a broom ring. Since L is the maximal ideal of the ring, we have shown that F(A, R)
is 3quasinormal. O
We present a similar example of an broom ring that is not quasinormal.
Example 5.1.11. Let Ao = [0, oo) C R, and A, = A2 = (, oo) C R, where each
is inversely ordered. Let A' = Ao and let A2 be the disjoint union A II A2 in
order to obtain a root system A = Al A2 with the induced ordering which we now
describe. That is, r < s if and only if either r, s E Ai for j = 0, 1 or 2 and r < s
in the inversely ordered real numbers; or if r E A,,s E Ao; or if r E A2,s E Ao.
Explicitly, r I s if r E Ai and s E A2.
We endow A with a partial binary operation. Let rj, sj E A, for j = 0, 1, 2.
Let (*)j denote the sum in parentheses as the usual sum of real numbers residing in
AI; the mark "" signifies that the sum is undefined.
+ ro 80 1r s81 2 82
ro (2ro)o (0o+ro)o (ri +ro)i (a +ro)i (r2 + )2 (82 + r)2
so (ro + oo) (20o)o (r +o)1o (l + 8o) (12 +o)2 (s +so)2
rt (ro+rl)l (so+r1) (2r)i (I1 + rl)l 
1I (ro+sl) (so+Ish) (r1 +ai (2ai) 
r2 (ro+r)2 8(o+r2)2 (2a2)2 (2 +,2)2
2 (tro+ 2)2 (0+8)2 (r2 +2)2 (202)2
Let Co = Ao, C = A0 II A1 and C2 = Ao I A2. Then the minimal prime ideals of
F(A,R) are of the form Qj = {u E F(A,R) : supp(u) C A \ Cj} for j = 1,2; the
ideal Qo, is a prime ideal by Proposition 3.3.15. Now, Qo = lex(F(A,R)) = Qi VQ2
and so we know that ir(F(A, R)) = 2. Therefore, Proposition 5.1.9 shows that the
ring is a broom ring. Since Qo is not the maximal ideal of the ring, we have shown
that F(A, R) is not quasinormal. ]
5.2 (B. m)Boundarv Conditions
Definition 5.2.1. Let m be an integer greater than 1 and let {Uj} =1 be a family of
m pairwise disjoint cozerosets of a topological space X. The subspace fl= dlx(Uj)
is called an mboundary in X. Let B be a topological property. We say that a space
X satisfies the (B, m)boundary condition if every mboundary in X has property
B.
In [Lal, 3.5], Larson proves that if X is completely regular, every point of X
is a Gapoint, and C(X) is quasinormal, then X satisfies the (discrete, 2)boundary
condition. This result is improved in [La3, 3.5] to say that if X is normal and for
every p E jX \ vX, the Iideal OP is prime then C(X) is quasinormal if and only if
X has the (finite, 2)boundary condition. Here we refine this theorem by removing
the restriction on the points of the corona.
First, a lemma, extending [La3, 3.1], which we henceforth refer to as "Larson's
Lemma":
Lemma 5.2.2. Let X be normal and let {gj} l C C(X)+ be a family of pairwise
disjoint functions. Define Ym = nLm= clx(coz(gj)). Then Eti gf = rE M,.
Proof: Each function in =1 gj must vanish on Y,, hence im1 g C nev M,.
For the reverse inclusion, we use recursion. The proof of the base case of m = 2 is
in [La3, 3.1].
Let f E nrYn Mv and define
f (x) = f A 1 if E cl(coz(gl)),
o0 if xE n7,=2l(coz(g)).
Since X is normal and fi is defined on a closed set, the function has a continuous
extension, J1 E C(X). Then fAlf1 E gjL and ft E n{Mp: p n.=2 cd(coz(gj))}.
Recursively define a function fk to be the continuous extension of
f() = I^0i A 1 if xE cl(coz(gk))
t0 if x E nfk+l (coz(g))
Then fk A 1 e E g and f E n{M, : p E ni'k+1, c(coz(gj))}. In particular, by
the base case, we have that fm2 E ({Mp : p E n m,,l1 oz(g))} = g9m1 + g.m
But then fn3 A 1 fm2 E 9gL2 implies that 3 A 1 E j m2 g and therefore
we have that m3 = (7m3 A 1)(7m3 V 1) E r,2g . Thus, by recursion, we
deduce that f1 E E, gj Hence fAl = (fAlf1 )+1, E m=gf and therefore,
f = (f A )(f V 1) E Em=! gi, as desired. *
Let X be normal and let Y,, as above, be given. The set Ym is Cembedded
in X by [GJ, 3D]. Thus we have a surjective ring homomorphism p : C(X) + C(Yn)
given by restriction of functions. The kernel of the homomorphism is
m
K = {f E C(X): Ym C Zx(f)} = N M= gf,
yEYm j=l
by Larson's Lemma. Thus by the First Isomorphism Theorem, it follows that
C(Ym) C(X)/K = C(X)/l gf.
j=1
We utilize the onetoone correspondence between the prime ideals of C(Y,) and
prime ideals of C(X) which contain K, the kernel of p.

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PRIME IDEALS IN RINGS OF CONTINUOUS FUNCTIONS By CHAWNE MONIQUE KIMBER A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1999
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This work is dedicated to those women who preceded me and to those who are yet to follow.
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ACKNOWLEDGMENTS First and foremost, I express my wholehearted gratitude to my advisor, Jorge Martinez. In the past few years, his humane guidance has helped me to achieve so very much, in fact, more than I would ever have hoped. I follow his example both in becoming a mathematician and a caring teacher, and in the enjoyment of the finer things in life like wine, cheese, and chocolate. Also, sincere thanks go to my committee members: Richard Crew, for show ing me some algebra; Alexander Dranishnikov, for teaching me a heap of topology; Scott McCullough, for introducing me to real analysis (back when we were both much younger); and Mildred HillLubin, for expanding my worldview through lit erature. Cheers and warm hugs to my friends, neighbors, and family, especially to the immediate: Johnnie, Charles, Frances, Maribell, Chinene, Jean, and the inimitable Poopygirl, Cei lll
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TABLE OF CONTENTS ACKNOWLEDGMENTS ABSTRACT CHAPTERS 1 PRELIMINARJES . . . . . . . 1.1 History . . . . . . . . 1.2 LatticeOrdered Groups . 1.3 /llings .......... 1.4 llings of Continuous Functions 1.5 Approaches . 2 CHARACTERS .............. 2.1 Hahn Groups .......... 2.2 Lex Kernels and Ramification . 2.3 Rank .............. 2.4 Rank via z#Irreducible Surjections 2.5 Prime Character . . . . 2.6 Filet Character . . . . . 3 GENERALIZED SEMIGROUP RJNGS 3.1 Specially Multiplicative /llings 3.2 rSystems and Systems . . 3.3 /Systems . . . . . .. 3.4 Survaluation lling and n th Root Closed Conditions 4 RAMIFIED PRIME IDEALS ............ 4.1 Ramified Points . . . . . . . 4.2 Ramified G 6 points ........... 4.3 Ramification via CEmbedded Subspaces 5 mQUASINORMAL /RINGS .......... 5.1 Definitions . . . . . . . . 5.2 (B, m) Boundary Conditions ..... 5.3 (3X, mQuasinormal and SV Conditions REFERENCES ....... BIOGRAPHICAL SKETCH. iv iii V 1 1 3 7 10 15 18 18 20 26 32 36 43 46 46 52 58 66 75 75 79 84 88 88 94 107 110 113
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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PRJME IDEALS IN RJNGS OF CONTINUOUS FUNCTIONS By Chairman: Jorge Martinez Chawne Monique Kimber May 1999 Major Department: Mathematics Given a completely regular topological space X, we wish to determine the order structure of < Spec ( C ( X)), >, the root system of prime ideals of the ring of realvalued continuous functions on X; and vice versa. We present four approaches which give partial solutions to these problems. First, we define three measures on< Spec+(G), ~>, the set of prime subgroups of a latticeordered group, which determine some arithmetic properties of the group, and vice versa. Second, given any root system, we construct a generalized semigroup ring 'R which is a commutative semiprime fring such that < f('R), ~>, its root system of values, is orderisomorphic to the given root system. Then we characterize those nonisolated G 0 points whose corresponding maximal ideal is the sum of the minimal prime ideals it contains. Finally, we characterize those spaces X for which C(X) has the property that the sum of any m minimal prime ideals is a maximal ideal or the entire ring. V
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CHAPTER 1 PRELIMINARIES The focus of this dissertation is the order structure of < Spec(C(X)), ~~;>, the spectrum of prime ideals of the ring C(X) of realvalued continuous functions on a topological space, X. To start, we informally present the history and give motivation for the discussion herein. We then review some essentials about latticeordered groups, /rings, and rings of continuous functions in detail and then formally indicate the manner in which this thesis proceeds. 1.1 History Our history begins with the independent research by Cech and Stone in 1937 (see the papers [Ce) and [St]), in which they describe a compactification {JX of a topological space X which has the property that every realvalued continuous func tion on X extends to a continuous function on {JX. Further, via {JX, they establish correspondences between the topological structure of X and certain algebraic prop erties of its ring C(X) of realvalued continuous functions under pointwise addition and multiplication. For instance, Stone shows that the maximal ideals of the sub ring C*(X) of bounded functions are in onetoone correspondence with points of {JX. The map p 1t M*P = {! E C*(X) : /(p) = O} witnesses this correspon dence and is a homeomorphism of topological spaces when the set of maximal ideals of the ring is endowed with the hullkernel (Zariski) topology. In particular, this shows that for compact Hausdorff spaces X, Y, we have that X ~ Y if and only if C(X) C(Y). The next significant result came in 1939 when Gelfand and Kolo mogoroff proved in [GK) that the maximal ideals of C(X) are exactly those of the 1
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2 form MP = {f E C(X) : p E cl13xZ(f)}, where Z(f) = {p E X : f(p) = O} and p E {3X. In the 1950's, Gillman, Henriksen, Jerison and Kohls began a formal in vestigation of topological/algebraic correspondences of this form. The elementary techniques and results are recorded in the text Rings of Continuous Functions, [GJ]. Concerning prime ideals, in [GJ, 14.3c] we learn that the prime ideals in C(X), con taining a given prime ideal, form a chain and in [GJ, 4J], it is shown that the topology on X is closed under countable intersections (i.e., X is a Pspace) if and only if every prime ideal of C(X) is maximal. More generally, in [GJ, 14.25] we find that X has the property that every bounded continuous function on set of the form X \ Z(f), for some f E C(X), extends to a continuous function on X (that is, Xis an Fspace) if and only if every maximal ideal of C(X) contains a unique minimal prime ideal. Knowing these three facts, we can describe the graph of the prime ideal spectrum of C ( X) in each case, where vertices are prime ideals and edges indicate setinclusion. In the Fspace situation, the graph is a disjoint set of strands (one for each point of {3X) with no branching; a Pspace yields a graph consisting solely of vertices (one for each point of {3X). It is from these topological characterizations of the graphical structure of the spectrum of prime ideals of C(X) that we formulate our questions. Roughly speaking, we wish to know: Is it possible to determine the order structure (under inclusion) of the prime ideal spectrum of the ring of continuous functions of a given topological space? Conversely, given a graph, is it possible to construct a topological space such that the given graph, in some sense, determines the structure of the spectrum of its ring of continuous functions?
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3 It turns out that both of these are rather ambitious pursuits and the ques tions must be refined before we can approach them. Extending the knowledge of properties of the prime ideals of C(X), Kohls published a series of papers ([Kl), [K2], and [K3]) in 1957. In [K2], he addresses the properties of chains of prime ideals of C(X). First he shows that the quotient ring C(X)/ Pis totallyordered for every prime ideal P and concludes that the prime ideals of the quotient ring form a chain. Second, it is demonstrated that if Pis nonmaximal, then the chain of prime ideals in C ( X) / P contains an 1/l set ( that is, a totally ordered set E such that for every pair of disjoint countable subsets A, B C E such that A < B, there exists c EE such that a< c and c < b for every a EA and every b EB). Hence, the chain of prime ideals contains at least 2N 1 primes. We may thus immediately reduce the second of our questions to only consider those graphs for which each nontrivial edge passes through an 1/l set of vertices. The facts presented in the next three sections show us that the class of graphs to consider can be further reduced. In this dissertation, we continue to refine the questions and present four perspectives ranging from the very general to the very specific which give partial results. In order to properly introduce these approaches, we must recall some facts and constructs which are fundamental to the ensuing investigation. 1.2 LatticeOrdered Groups Let ( L, ~) be a partially ordered set. If a, b E L are incomparable, then we write a II b. L is totally ordered if any two elements are comparable. We say that L is a lattice if any two elements a, b E L have a least upper bound and a greatest lower bound, denoted a V b and a I\ b, respectively. A lattice L is distributive if a I\ (b V c) = (a Vb) I\ (a V c), and dually for all a, b, c EL. A group (G, +, 0, ~) with partial order is a latticeordered group (hence forth, group) if it is a lattice and if g h implies that c+g c+h and g+c h+c
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4 for all c E G. The majority of the groups we consider are abelian, so the additive notation here is for convenience. It is important to note that any group is torsion free [D, 3.5] and its lattice is distributive [D, 3.17]. A {real) vector lattice is an group G which is also an JRvector space such that rg 0 for all positive g E G and for all positive r E JR. By a+ we mean the set of elements g E G such that g 0. Each element of G may be written as a difference of elements of a+ : let g+ = g VO and g= (g) V 0, then g = g+ g . This follows from the fact that g+ I\ g = 0. The absolute value of an element is given by jgj = g+ + g. In general, we say that a pair of elements g, h E Gare disjoint if g I\ h = 0. We will write g h if ng < h for all n EN. An homomorphism is a group homomorphism that also preserves the lattice structure. An subgroup H of an group G is a subgroup which is also a sublattice of (G, ~). We call an subgroup convex if O g h EH implies that g EH G(S) denotes the convex subgroup of G generated by the set S G. When S = {g}, we write G(g). In fact, G(g) = {h E G : 3n E N, lhl njgj}. In the special case that G (g) = G, we call g a strong (order) unit. Let
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5 ..l, and V~ 1 Hi = (UiEI Hi)11, where { HihEI s.Jl( G) and J is any indexing set, we have that q:l(G) is a Boolean lattice, by (D, 13.7]. The convex subgroups of greatest interest to us are the prime subgroups of G. These are the subgroups HE
PAGE 11
6 have P = U{g 1. : g (j. P}. This implies O(Q) def n{P E Min+(G) : P Q} is the set U{g1.: g (j. Q}, for a prime subgroup Q G, by [BKW, 3.4.12). A basis for an lgroup G is a maximal pairwise disjoint set {9ihe1 a+ such that for each i E J the set {g E a+ : g 9i} is totally ordered The following is Conrad's Finite Basis Theorem presented as [D, 46.12) and [C, 2.47). It will figure in our discussion in the next chapter. Theorem 1.2.1. Let G be an l group. The following are equivalent: 1. G has a finite basis. 2. Min+ ( G) is finite. S. s.JJ( G) is finite. 4. There is a finite upper bound on the number of pairwise disjoint elements of G. 5. There is a finite upper bound on the number of elements of strictly increasing chains of proper polars. Another application of Zorn's Lemma establishes the existence of convex subgroups which are maximal with respect to not containing a fixed element g E G. Any such subgroup is generally termed a regular subgroup and specifically called a value of g. The set of all regular subgroups of G is usually represented by f(G). Regular subgroups are prime, by [D, 10.4], and a prime subgroup is precisely a convex subgroup which is an intersection of a chain of regular subgroups, [D, 10.8]. In particular, the minimal prime subgroups of G correspond to the maximal chains in r(G). For these reasons, we call the root system given by r(G) the skeleton of Spec+(G). By convention, we view r(G) as a partiallyordered indexing set r whose
PAGE 12
7 elements are denoted by lower case Greek letters and then represent the regular subgroups by Vy for 'YE r. Topologize Spec+ ( G) using the hullkernel (or Zariski) topology whose open base is given by U(g) = {P E Spec+(G) : g ff. P} for all g E G. In this topology, Spec+(G) is Hausdorff if and only if Spec+(G)=Min+(G) by [CM, 1.4); on the other hand, in the subspace topology, Min+(G) is always Hausdorff and U(g) n Min+(G) is both open and closed for every g E G. The space Spec+ ( G) is compact if and only if G has a strong unit, by [CM, 1.3); it is demonstrated in [CM, 2.2) that Min+(G) is compact if and only if G is complemented, that is, if and only if for every g E G+ there is an h E G+ such that g A h = 0 and g V h is a weak unit. 1.3 /Rings Let (R, +, :::;) be a ring whose underlying group is an group and satisfies the relations re :::; sc and er :::; cs whenever r :::; s and c 2:: 0. Such a ring is a lattice ordered ring (abbreviated lring). Han ring R also satisfies ca Ab= ac Ab= 0 whenever a I\ b = 0 and c 2:: 0, then R is called an fring. The following is found in [BKW, 9.1.2): Theorem 1.3.1. Let R be an lring, then the following are equivalent: 1. R is an fring. 2. Every polar in R is an ideal. S. Every minimal prime subgroup of R is an ideal. It is not difficult to verify that every ring which is isomorphic to a subdi rect product of totallyordered rings ( with coordinatewise operations) is an fring. In [BP), it is shown that the converse of this statement holds when we assume the Axiom of Choice (abbreviated, AC). Since we routinely apply AC, let us formally
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8 state that we will work within the axioms of ZFC. Then we may use this equiv alent definition of an fring R in order to obtain this list of arithmetic properties given in [BKW, 9.1.10], for a, b, c ER: 1. If c > 0 then c(a Vb)= ca V cb and (a V b)c = ac V be. 2. If c > 0 then c(a I\ b) = ca I\ cb and (a I\ b)c = ac I\ be. 3. lal lbl = labl 4. If a I\ b = 0 then ab = 0. 5. a 2 0. An ideal of an ring R is an ideal which is a convex subgroup of R. We call an ideal a prime ideal if it is also a prime ideal. Let Spec( R) denote the space of all prime ideals of R in the hullkernel toplogy. Let Max(R) and Min(R) denote the subspaces of maximal and minimal prime ideals, respectively. By property (4) above, we see that prime ideals of an fring are prime subgroups, hence, as in the case of groups, the subset Spect(R) of prime ideals forms a root system. Denote the subspaces of maximal and minimal prime ideals by Maxt(R) and Mint(R), respectively. We call a commutative ring semiprime if it contains no nonzero nilpotent elements. In the case of commutative !rings, we have [BKW, 9.3.1]: Theorem 1.3.2. Let R be a commutative !ring, then the following are equivalent: 1. R is semiprime. 2. For any a, b ER, we have that lal /\ lbl = 0 if and only if ab= 0. 3. Every polar of R is an ideal which is an intersection of prime ideals.
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9 4Min,(R) = Min(R) 5. R is isomorphic to a subdirect product of totallyordered integral domains. We say that an ring R with multiplicative identity, 1, has the bounded inversion property if a 1 implies that a is a multiplicative unit. By [HIJo, 1.1], a commutative /ring R with 1 has the bounded inversion property if and only if Max,(R) = Max(R). Let A be any commutative ring. Then, in the hullkernel topology, Min(A) is a Hausdorff space with a base of clopen sets. If A is a semiprime ring, then Spec(A) is Hausdorff if and only if Min(A) =Max(A); this occurs if and only if A is von Neumann regular (or absolutely flat,) i.e., for every a E A there exists b E A such that a= a 2 b), see [AM, p. 35]. In [HJ] it is demonstrated that if A is a semiprime /ring, then Min(A) is compact if and only if A is complemented (i.e., for every a EA there exists b EA such that ab= 0 and a+ bis not a zerodivisor) Max(A) is compact for any commutative ring A with identity, and if A is a commutative fring with identity which has the bounded inversion property, then the subspace is Hausdorff, see [HJo, 2.3]. Let A be a commutative ring with identity and PE Spec(A). Define O(P) = {a EA: 3b (j. P, ab= O}. If A is also a semiprime fring, then this is the same as the subgroup O(P) defined in the previous section. Recall that the localization of A at Pis the subring, Ap, of the classical ring of quotients of A/O(P) consisting of the elements whose denomi nator is not in P/O(P). (For a review of this construction and general facts about localizations, see [AM] or [G]). It is the case that Ap is a local ring whose unique maximal ideal is generated by P/O(P) and there is a onetoone correspondence between prime ideals of Ap and the prime ideals Q of A such that O(P) Q P.
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10 Thus, if A is also an /ring with bounded inversion, then by the root system struc ture of Spec,(A), we have that AM A/O(M) since the quotient ring is already local with unique maximal ideal M/O(M). 1.4 Rings of Continuous Functions Let X be a Hausdorff topological space. X is called completely regular (or Tychonoff) if for every closed set A~ X and x EX\ A, there exists a realvalued continuous function on X such that J(x) = 1 and J(A) = {O}. Unless otherwise stated, we assume that all spaces are completely regular. Let C(X) denote the set of realvalued continuous functions on a space X. Under the operations of pointwise addition and multiplication, C(X) is a semiprime ring. Order the ring via: / g if and only if f(x) g(x) for all x E X. This ordering gives an /ring structure such that C(X) has the bounded inversion property. Let C*(X) denote the convex lsubring of bounded functions. The zeroset of/, is the set Z(J) = {x EX : f(x) = 0}. The complement, coz(J) = X \ Z(J), is the cozeroset off. By (GJ, 3.6], a Hausdorff space X is completely regular if and only if its topology is the same as the weak topology generated by C(X). Equivalently, the set of all zerosets, Z(X), is a base for the closed sets of such a space, (GJ, 3.2]. Sets A, B X are completely separated if there exists f E C(X) such that J(A) = {O} and J(B) = {l}. If for every f E C(A) there exists TE C(X) such that jjA = f, then we say that A is Cembeddedin X. Likewise, A is C*embeddedin X if bounded continuous functions on A extend to bounded continuous functions on X. These embedding properties are characterized by complete separation of particular subsets, as follows: 1. Urysohn Extension Theorem (GJ, 1.17]: A~ Xis C*embedded in X if and only if any two completely separated sets in A are completely separated in X.
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11 2. A C* embedded set is Cembedded if and only if it is completely separated from every zeroset disjoint from it, [GJ, 1.18]. Recall that a Hausdorff topological space X is normal if any two disjoint closed sets are separated by disjoint open sets. Assuming this stronger separation axiom, the results listed above give rise to the theorem stated in [GJ, 3D], in which the equivalence of the first two statements is known as Urysohn's Lemma. Theorem 1.4.1. Let X be Hausdorff. The following are equivalent: 1. X is normal. 2. Any two disjoint closed sets of X are completely separated. 9. Every closed set of X is C*embedded in X. 4. Every closed set of X is Cembedded in X. For many reasons, it is often preferable to work with compact spaces. The StoneCech compactification {3X is our compactification of choice, since {3X is char acterized by the property that it is (up to homeomorphism) the unique compact space in which X is dense and C* embedded. There are at least three different ways to construct {3X, we begin with the one based on ultrafilters, described in detail in Chapter 6 of [GJ], which we now summarize. Let X be a completely regular space, let C be a subset of the power set of X and :F C. :Fis a Cfilter if 0 rt :F, it is closed under finite intersections and if for every F E :F, the fact that F F' E C implies that F' E :F. If :Fis a Z(X) filter, then :F is also called a zfilter. A maximal filter is an ultrafilter; similarly, a zultrafilter is a maximal z:6.lter. Let {3X be the set of all zultrafilters on X which we index by { AP : p E /3X}. A closed base for the topology on {3X is given by sets of the form Z = {p E {3X: Z E AP}, for Z E Z(X). Let p E {3X and define MP= {f E C(X) : p E clpxZ(f)}.
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12 The theorem of Gelfand and Kolmogoroff [GK] is stated simply as: Theorem 1.4.2. For a completely regular space X, the set Max(C(X)) is given by {MP: p E ,BX}. In fact, this result gives rise to a homeomorphism of ,BX with Max(C(X)). That is, since the sets Z[MP] = { Z(f) : f E MP} are precisely the zultrafilters on X, by [GJ, 2.5], and Theorem 1.4.2 shows that the map pi+ Z[MP] is the desired correspondence. If p E X, then we will write Mp and, in this case, the maximal ideal and corresponding zultrafilter are called fixed. Otherwise, a maximal ideal and its corresponding zultrafilter is called free. It is evident that X is compact if and only if every maximal ideal of C(X) is fixed. Maximal ideals are also classified by the residue field C(X)/MP. Identifying the constant functions with their constant, we see that these fields always contain a copy of JR. We call a maximal ideal real if the field is exactly R; otherwise, the maximal ideal is called hyperreal. This concept is the basis for considering the Hewitt realcompactification of X. Denoted vX, it is the smallest subspace of ,BX in which X is dense and such that every maximal ideal of C(vX) is real. In fact, by [GJ, 8.5], vX is the largest subspace of ,BX in which X is Cembedded. With these facts about the maximal ideals firmly in place, we now proceed to consider the nonmaximal prime ideals. We know that every prime ideal of C(X) is convex, by [GJ, 5.5]; so we deduce that Spec(C(X)) is a root system. In order to understand this root system, we are required to consider the properties of other ideals. For instance, for p E ,BX, the ideals of the form QP = O(MP) = {J E C(X) : dpxZ(f) is a neighborhood of p} are of paramount interest when examining the prime ideals of C ( X). One reason is given in [GJ, 7.15]:
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13 Theorem 1.4.3. Every prime ideal P in C(X) contains OP for a unique p E {3X and MP is the unique maximal ideal containing P. If OP is prime, then we call pan Fpoint. If X has the property that OP is a prime ideal for every p E X, then we call X an Fspace. We see that in this case, the graph of Spec(C(X)) consists of a set of strands with no branches. Note [GJ, 14.25]: Theorem 1.4.4. Let X be completely regular. The following are equivalent: 1. X is an F space. 2. {3X is an Fspace. 9. The prime ideals contained in any given maximal ideal form a chain. 4. Every cozeroset of X is C* embedded. 5. Any two disjoint cozerosets of X are completely separated. 6. Every ideal of C(X) is convex. 7. Every finitely generated ideal of C(X) is principal {i.e., C(X) is Bezout) A special case of an Fpoint is when Op = Mp and we call pa Ppoint if this occurs. Call X a Pspace if every point of X is a Ppoint. In this case, the spectrum of C(X) consists only of vertices. Equivalent definitions of Pspace are presented in [GJ, 14.29] and are recorded below. First, recall that an ideal J of C(X) is called a zideal if f E J and Z(/) = Z(g) implies that g E J. It is immediate from the definitions that MP and OP are zideals for all p E {3X. Note that not all prime ideals are zideals; however, the following says that this is the case in a von Neumann regular ring.
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Theorem 1.4.5. Let X be completely regular. The following are equivalent: 1. X is an Pspace. 2. vX is an Pspace. S Every prime ideal of C(X) is maximal ,4.. Every cozeroset of X is Cembedded. 5. For each f E C(X), the zeroset Z(f) is open. 6. Every ideal of C(X) is a zideal. 14 7. For every f E C(X), there exists g E C(X) such that f = g/2 {that is, C(X) is von Neumann regular). We now recall the definitions of other types of spaces which are useful to us. X is basically disconnected if the closure of any cozeroset is clopen. X is extremally disconnected if any open set has open closure. Discrete spaces are extremally dis connected; extremally disconnected spaces are basically disconnected and all such spaces are Fspaces by [GJ, 14N.4). Every Pspace is basically disconnected by [GJ, 4K.7). A space is a quasiF space if every dense cozeroset is C*embedded. Clearly, from [GJ, 14.25), we see that every Fspace is quasiF. The converses of the pre ceding statements do not hold. That is, these are distinct classes of spaces, as we now illustrate. Example 1.4.6. Consider the following spaces: 1. Let U be a free ultrafilter on N. Let = N U {a}, in which points of N are isolated and neighborhoods of a are of the form U U {a}, where U EU. Then~ is an extremally disconnected subspace of (3N, but not a Pspace. In particular, O
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15 2. Let D be an uncountable set. Let )..D = D U {)..}, where points of D are isolated and a neighborhood of ).. is given by any cocountable set containing it. Then )..Dis basically disconnected, but not extremally disconnected by [GJ, 4N.3]. Moreover, the topological sum X = >.D II~ is basically disconnected, but neither extremally disconnected nor a Pspace, by (GJ, 4N.4]. 3. The corona, {JN \ N is a quasiF space which is an Fspace, yet not basically disconnected; see [GJ, 6W.3, 140]. 1.5 Approaches Starting as generally as possible in Chapter 2, we define three cardinalvalued characters on the spectrum of prime subgroups of an group. The value of each measure determines a portion of the arithmetic and/or polar structure of the group, and vice versa. For instance, we define the prime character, 1r( G) of an group, G to be the least cardinal K such that for any family {Qa}a<" Min+(G), of distinct minimal prime subgroups, we have that Va<" Q 0 is the smallest convex subgroup of G containing all the elements of Min+(G). Roughly speaking, it is a measure of the complexity of minimal paths in the graph of Spec+ ( G) between minimal prime subgroups. We will show that the measure being finite satisfies the following, where lex( G) denotes the smallest convex subgroup of G containing all the elements of Min+ ( G) : Proposition 1.5.1. Let G be an lgroup and m a positive integer. The following are equivalent: 1. 1r(G) = m < oo. 2. m is minimal with respect to the property that lex(G) = G(LJ; 1 a{) for any m pairwise disjoint positive elements, {a;}J!: 1 lex(G)+.
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16 S. m is minimal such that for any prime P lex(G), the chains of proper polars in P have length at most m 1. Chapter 3 is devoted to a discussion of the properties of F(fl, JR.), the gener alized semigroup ring of realvalued maps on a root system fl (which has a partially defined associative operation, +) each of whose support is the join of finitely many inversely wellordered sets. The ring structure on this group is introduced in [Cl] and [C2]; we endow this ring with an fring structure. In particular, we show that if (fl,+) is a root system such that each of the following holds: 1. + is associative (when it makes sense); 2. if a, /3 E fl are comparable, then a+ /3, /3 + a are defined; 3. if a< /3 and a+ 'Y, /3 + 'Y are defined, then a+ 'Y < /3 + 'Y and if 'Y + a, 'Y + /3 are defined then 'Y +a< 'Y + /3; 4. and if is maximal, then 8 + + o, + are defined and 8 + = + o = o for every o :::; then F(fl, JR.) is an fring if and only if 8 = a + /3 implies a, /3 2'.: o. And when this occurs, the fring is semiprime and satisfies the bounded inversion property. Moreover, by [CHH, 6.1), given any root system fl, one of these !rings has fl orderisomorphic to its root system of values. Thus, the second of our questions is answered in the class offrings on the level of skeletons. However, the solution to the second problem remains unclear in the smaller class of rings of continuous functions. To gain a modicum of clarity on the situation, we look to the work of Attilio LeDonne, published in 1977 in [Le), in which he addresses the incidence of branching in the graph of Spec(C(X)). He shows, for instance, that the root system branches at every prime zideal when X is a metric space. In [Le, ), LeDonne includes a result of DeMarco which states that there
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17 is branching at each Mp when X is a firstcountable space and pis nonisolated. In Chapter 4, we show that, for a nonisolated G 6 point of a completely regular space, there is branching at Mp if and only if X \ p is not c embedded in X. This result is then used to examine branching in Spec(C(X)) when Xis not necessarily firstcountable. Both of our questions are addressed in Chapter 5, in which we generalize a few of the results of Suzanne Larson on quasinormal /rings that are found in the series of papers [Lal], [La2], and [La3]. The semiprime commutative quasinormal /rings with identity are the ones having the property that the graph of the root system of prime ideals does not contain a subgraph of the form: (1.1) By (La3, 3.5] a normal space X has the property that C(X) is quasinormal if and only if cl(U) n cl(V) is a Pspace for any disjoint cozerosets U, V X Our generalizations similarly describe those normal spaces X for which Spec(C(X)) does not contain a subgraph of any of the following forms: n n (1.2) where n, k, ai, ... an are positive integers satisfying some specified conditions.
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CHAPTER 2 CHARACTERS We seek a collection of measures on root systems whose values will determine some portion of the structure of a latticeordered group. In this chapter we describe three such measures: rank, prime character, and filet character. The rank measures the width of a connected component of the spectrum, the prime character determines, roughly speaking, the complexity of minimal paths between minimal primes, and the filet character counts the maximum length of a chain of branching incidences. The first sections of this chapter are a review of two constructs essential to the discussion to follow. 2.1 Hahn Groups To begin, we recall a method of constructing examples of groups having a specified root system as the skeleton of its prime spectrum. Let 6.. be a root system and define V(6.., JR) = { v: 6.. JR: supp(v) has ACC}, where supp(v) = { 8 E 6..: v(8):/= O}. V(6.., JR) is an group under pointwise addition ordered by the relation: v > 0 if and only if v ( 8) > 0 for every maximal element 8 E supp(v). This group is called a Hahn group. In the paper of Conrad, Harvey and Holland [CHH], it is demonstrated that any abelian group can be embedded in a Hahn group of a more general description than we give here. Of interest to us is the subgroup of maps with finite support denoted by E(6.., JR) and the subgroup of maps whose support is the join of finitely many inversely wellordered sets, denoted by F(6.., JR). Clearly, E(6.., JR) F(6.., JR). 18
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19 The proof of the first statement of Proposition 2.1.1 is analogous to that of Theorem 6.1 in [CHH). This establishes that r(E(.6., R)), f(F(.6., R)) and .6. are isomorphic as partiallyordered sets. For the sake of completeness, we present an elementary proof of this fact for the case of F(.6., R), although the result is easily obtained from the theory of finitevalued groups. The proof is identical in the case of E(.6., R). Recall that an group is finitevalued if each element has only a finite number of values. A special value is a prime subgroup which is the unique value of an element. An group G is finitevalued if and only if every value of G is special and if and only if every element of G is a finite sum of pairwise disjoint special elements; for details, see [AF, 10.10). HG has a set S of special values such that Sis a filter and ns = { 0}, then G is called specialvalued. Proposition 2.1.1. Let .6. be a root system. For each 8 E .6. define = {f E F(.6., JR) : v(,) = 0 when 'Y 8}. Each is a special value. Further, every value of an element of F(.6., JR) is of the form for some 8. Thus, .6. is the skeleton ofSpec+(F(.6.,R)) and F(.6.,JR) is finitevalued Proof: Let 8 E .6. and let X6 E F( .6., JR) be the characteristic function on { 8}. Then X6 0. H 8 < 'Y, then O < X6 < v, a contradiction. H 8 = ,, then there exists a positive integer n such that O < X6 nv and hence x 6 E V by convexity, which is a contradiction. Thus V = Let v E F(.6., R)+ and let V be a value of v. Let D be the finite set of maximal elements of supp( v). Then the characteristic function xn is not in V; else,
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20 there exists an integer n such that O v n xv, a contradiction. Since V is prime and the set {x 6 : o E D} is pairwise disjoint, there exists a unique element o E D such that X6 V. By the above, we know that V Finally, since v we have that V = Vo, as desired. The final statement follows from [AF, 10.10] since we have shown that every value is special. 2.2 Lex Kernels and Ramification Throughout we will describe the location of a prime subgroup in the graph in reference to a designated convex subgroup, called the lex kernel of an group G and denoted by lex(G). It is the least convex subgroup containing all the minimal prime subgroups of G. It is always the case that lex(G) is a prime subgroup [D, 27.2] which is normal in G [D, 27.13]. The following is a summary of a part of the discussion of lex kernels in [D, ] and gives a description of the subgroup in terms of its generators. Proposition 2.2.1. Let G be an group and let C be a convex subgroup. The following are equivalent: 1. C = lex(G). 2. C is the least prime subgroup such that if O < g C then g > h for every hE C. S. C is the convex subgroup of G generated by {g E G: g II O}. 4. C = {O} U {g E G : 391, 92, E G, g II 91 II 92 II II Un II O}. 5. C is the convex subgroup of G generated by the nonunits of G. 6. C is prime and is the smallest among all convex subgroups of G which are comparable with every convex subgroup of G.
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21 7. C is the maximal convex subgroup of G such that lex( C) = C. 8. C is the supremum of the proper polars of G in the lattice of convex subgroups ofG. It is natural to now introduce a concept which we will discuss in more detail in Chapter 4. This is a generalization of a concept from [Le]. Let A be a commutative ring with identity and for each a EA, let Max(a) ={ME Max(A): a EM}. Recall that an ideal I of A is a zideal if a E J and Max(a) = Max(b) imply that b E J Definition 2.2.2. Let A be a commutative !ring with identity. A prime l.ideal P is ramified if it is the sum of the minimal prime ideals that it contains. A maximal l.ideal M is totally ramified if every prime zideal contained in M is ramified. A completely ramified ring is one in which every prime zideal is ramified. Graphically, a prime ideal P :5 A is ramified if and only if it is minimal or if the root system of prime ideals of A branches at P. We begin with the group characterization of ramification. It is the case that a ramified maximal ideal M of A is the lex kernel of the local /ring A/O(M). In order to discuss a proper lex kernel in an ring, A, we must operate inside a localization. Henceforth, we will obtain results for local rings and tacitly extend to the general case by referring to localizations. The following characterization of ramified maximal ideals is immediate from Proposition 2.2.1. Corollary 2.2.3. Let A be a commutative semiprime local !ring with identity and bounded inversion and let M be the maximal ideal. The following are equivalent: 1. M is ramified. 2. M is the convex subgroup of A generated by {f EA:/ II O}.
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22 3. M = {O} U {/EA: 3/1, /2, fn EA, f II /1 II /2 II 11 fn II 0}. ,4.. M is the convex subgroup of A generated by the set {/EA: 3g EA, g > 0, g I\ f = 0}. 5. M is the convex lsubgro'Up generated by the elements of A which are not order units. 6. M is the smallest among all convex subgroups of A which are comparable with every convex subgroup of A. 7. Mis the supremum of the proper polars of A in the lattice of convexlsubgroups of A. It is wellknown that the lex kernel of an group is a prime subgroup (see [D, 27.2]). We now show that the lex kernel of a commutative local semiprime /ring with identity is an ideal and then give conditions which guarantee that the lex kernel is a prime ideal. Let A be a commutative /ring with identity. Recall that an ideal J 5 A is pseudoprime if ab = 0 implies a E J or b E J. An ideal J 5 A is semiprime if a E J whenever a 2 E J. A is squareroot closed if for any 0 5 a E A, there exists 0 5 b E A such that a = b2. Let a, b E A, then A is nconvex if whenever 0 5 a 5 bn, there exists u E A such that a = bu. Proposition 2.2.4. Let A be a commutative semiprime local /ring with identity. Then lex(A) is a prime s'Ubgroup which is a pseudoprime ideal. If, in addition, A is squareroot closed, then lex(A) is a semiprime ideal. Proof: Let f E lex(A). Then there exists g > 0 such that f I\ g = 0. If af = 0 then af E lex(A); else, af I\ g = 0 and we conclude again that af E lex(A). Hence lex(A) is an ideal.
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23 Let N be the set of nonunits of A and recall that lex(A) = A(N). Let ab = 0. If a or b is 0, then there isThen by convexity, we see that a+, a E A(N). Hence a E A(N) and we have that the lex kernel is pseudoprime. Since any prime ideal is semiprime and the lex kernel is the sum of the minimal prime ideals, the lex kernel is semiprime, if A is also squareroot closed; see [HLMW, 2.12(d)]. Corollary 2.2.5. Let A be a commutative semiprime local fring with identity and bounded inversion and let M be the maximal ideal. M is ramified if and only if lex(A) is a zideal. Proof: Since the maximal ideal is the only zideal of a local fring, this is immediate. Corollary 2.2.6. If A is a commutative local 2convex semiprime / ring with iden tity which is squareroot closed, then the lex kernel of A is a prime ideal. Proof: By the remark after [La4, 4.2], under these hypotheses, we have that an ideal is a prime ideal if and only if it is pseudoprime and semiprime. For the remainder of this section, let G be an abelian latticeordered group. Recall the following for Han subgroup of G His rigid in G if for every h EH there is g E G such that h.1.1 = g.1.1_ It is shown in [CM, 2.3] that if His rigid in G then the contraction of minimal prime subgroups of G to minimal prime subgroups of H is a homeomorphism of minimal prime spaces. If H E ~( G), then H is very large in G if it is not contained in any minimal prime subgroup of G. It is shown in [CM] that if HE ~(G) then His very large in G if and only if His rigid in G. It turns out that ramification in a rigid subring indicates global ramification and vice versa. This is a direct consequence of the lex kernel correspondence demonstrated below.
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24 We will also need the following facts (see [BKW, 2.4.7, 2.5.8]): Proposition 2.2.7. Let HE t(G). 1. The contraction map from the set of prime subgroups of G not containing H to the set of prime subgroups of H is an orderpreserving bijection. 2. IfV is a value of h EH in G, then V HV n H is a bijection between the set of values of h in G and the set of values of h in H. Proposition 2.2.8. Let H G be a convex subgroup. Assume that u E H is a weak unit of H and a weak unit of G. Let V be a value of u in G. Then we have that V n H = lex(H) if and only if V = lex(G). Proof: Assume that H P E Min+ ( G) then u E P and we have that u.1 c/:. P by [AF, 1.2.11). This is a contradiction since u.1 = 0 E P. Thus His rigid in G since it is a convex subgroup which is very large in G. Assume that V n H is the lex kernel of H. Then V n H is the least convex subgroup of H containing all the minimal prime subgroups of H. Since His rigid in G, by the bijection given in the first part of Proposition 2.2.7, all the minimal prime subgroups of G are contained in V, and V is the least such convex subgroup of G. That is, if W V also contains the minimal prime subgroups of G, then W n H is a convex subgroup of H containing all the minimal prime subgroups of Hand hence W n H = V n H. But this says that W n HE r(H) is a value of u and hence, WE r(G) is a value of u Therefore V = W If Vis the lex kernel of G then VnH contains all the minimal prime subgroups of H. Thus the lex kernel of H is contained in V n H. Let P c V n H be a prime subgroup of H containing all the minimal prime subgroups of H. Then by Proposition 2.2. 7, there exists a prime convex subgroup Q G not containing H such that P = Q n H and since we have a rigid embedding, Q contains all the
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25 minimal prime subgroups of G. Hence, Q = V, P = V n H and V n H is the lex kernel of H. Let A be a commutative semiprime fring with identity and bounded in version. If M E Max(A) then AM is semiprime with bounded inversion. This is a result of the wellknown facts that the homomorphic image of an !ring with bounded inversion has bounded inversion and that AM A/O(M); see the proof of [La3, 2.7]. Since we must localize an fring in order to have a proper lex kernel, the following allows application of Proposition 2.2.8 to frings. Proposition 2.2.9. Let B be a commutative semiprime !ring with identity and let A be a rigid convex fsubring of B. If M E Maxt(A) is such that M = N n A for some NE Maxt(B), then AM is a rigid convex fsubring of BN, Proof: Recall that AM~ A/O(M) and BN ~ B/O(N). Define a map: A BN by a~ a+O(N). This map is an !ring homomorphism. We show that the image is convex in BN and that the kernel is O(M). For a EA, let O b+O(N) a+O(N). Then there exists n E O(N)+ such that O b a+n. If bn 0 then O < b n and hence b E O ( N) since O ( N) is convex in B. Thus we may assume that O bn a. Then bn EA and b+ O(N) = bn + O(N) Elm(). Therefore the image of is convex in B N. The kernel of is O(N) nA. It is easy to show that O(N) nA O(M) since M = N n A. For the reverse inclusion, assume that a E A and a O(N). Since O(N) is the intersection of the minimal prime ideals of B contained in N, there exists P EMin(B) such that P ~Nanda P. By the rigidity of A in B, PnA EMin(A), and therefore a O(M). We now have that Ker() = O(N) n A = O(M) and therefore AM is a convex fsubring of BN. Since AM contains the identity element of B N, AM is very large in B N. For rigidity, we need only recall that very large convex embeddings are rigid, [CM].
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26 Corollary 2.2.10. Let B be a commutative semiprime !ring with identity. Let A be a rigid convex fsubring of B. Let M E Maxt(A) be such that M = N n A for some NE Maxt(B). Then M is ramified in A if and only if N is ramified in B. Let A* denote the fsubringofbounded elements of the commutativesemiprime fring A with identity. Note that A* is convex and rigid in A. In his dissertation [Wo], Woodward proves the following fact: Theorem 2.2.11. Let A be a semiprime !ring with identity and bounded inver sion. Let M be a maximal ideal of A and let M be the unique value of A* containing Mn A*. The map M M gives a homeomorphism between Max(A) and Max(A*). That is, Max{A*) is the subspace consisting of values of l in A. In particular, if M is a real maximal ideal of A, then Mn A E Max(A*). Corollary 2.2.12. Let A be a commutative semiprime !ring with identity satisfy ing the bounded inversion property. Let ME Max(A) be real. Then M is ramified if and only if M = M n A is ramified in A. 2.3 Rank The first character on Spec+ ( G) that we consider is simply one which counts the minimal prime subgroups contained in a convex subgroup. Definition 2.3.1. The rank, rkG(H) of a convex subgroup H G is the cardi nality of the set of minimal prime subgroups of G contained in H. If that cardinal is not finite, then we will say that H has infinite rank; we may choose to specify the cardinal when its value is of significance in a discussion. If H is a minimal prime subgroup of G, then we define rkG(H) = 0. This is a variation of the following definition given in [HLMW): Let A be a commutative !ring with identity and M a maximal ideal of A. The rank of M, denoted rkA(M), is the cardinality of the subspace of minimal prime ideals of A
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27 contained in M. By convention, if the rank of Mis infinite and we don't necessarily care about the exact cardinality, we write rkA(M) = oo. The rank of a point p EX, rkx(p), is the rank of Mp. The rank of the /ring A is the supremum of the ranks of the maximal ideals of A, when it exists; the rank of a space X is the rank of C(X). We begin with illustrations of the extremal values of ranks. An group is semiprojectable if for any g, h E G+, (g I\ h)1= G(g1U b1). In [BKW, 7.5 1], it is proved that G is semiprojectable if and only if each prime subgroup contains a unique minimal prime subgroup, which is equivalent to rka(P) 1 for every PE Spec+(G). Thus, it is evident that a space Xis an Fspace if and only if C(X) is semiprojectable which is equivalent to rk(C(X)) 1. In particular, Xis a Pspace if and only if C(X) is von Neumann regular, which is equivalent to rk(C(X)) = 0. The onepoint compactification of the natural numbers, aN, is an example of a space for which C(X) has infinite rank, [GJ, 14G]. In fact, if a is the point at infinity, then the maximal ideal corresponding to a contains 2' minimal prime ideals one for each free ultrafilter on N. Moreover, by [HJ, 4.8], this subspace of minimal prime ideals is homeomorphic to the corona, {JN \ N. Proposition 2.3.3 describes a general situation in which we have infinite rank. We recall some definitions. From [LZ, 39.1]: let G be a vector lattice, v Ea+, and let {gn}~=l G be a sequence. We say that the sequence converges relatively uniformly to g E G along the Te!]'Ulator v, and write gn g, if for every > 0 there exists NE> 0 such that for all n NE, we have that jg gnl V. The sequence is relatively uniformly Cauchy with respect to v if for every > 0 there exists NE > 0 such that for all n, m NE, we have that jgm gnl V. G is called uniformly complete if for every v E a+, every sequence which is relatively uniformly Cauchy with respect to v relatively uniformly converges along the regulator v.
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28 Lemma 2.3.2. Let G be a uni/ ormly complete vector lattice with weak order unit u Ea + For any set {g;};ew, there exists g E G such that g = n;ewYT Proposition 2.3.3. Let G be a uniformly complete complemented vector lattice with weak order unit u E a+. If for some Q E Spec+(G) we have rkG(Q) ;?: w, then Q contains at least 2' minimal prime subgroups Proof: Note that, by [CM, 2.2], Min+(G) is compact since G is complemented. Let P = { Pn}neN be a countably infinite set of minimal prime subgroups which are contained in Q which is discrete in the hullkernel topology on Min+(G). We first show that P is c embedded in Min+ ( G) and conclude that the minimal prime space contains an homeomorphic copy of {IN. Then we describe the elements of Min+ ( G) that correspond to the points in this copy of /JN \ N. Let A, B P be completely separated in P and index them by I, J N as A = { Ai : i E J} and B = { B; : j E J}. Fix i E J and let Ai E A. For each B; E B, Let 'i ; E A+\ B; and b; E Bf\ Ai. Then E n;eJ U(b;J def Ki and B U;eJ U(D.i ; ) = U(E;eJ 2i('i; I\ u)) def Li. Then Ki, Li are disjoint closed sets in Min+(G). For each i E J, generate the disjoint pair Ki, Li. Then A~ cl(UieJKi) def K and B nie1Li def L. By Urysohn's Lemma, the disjoint closed sets K, L are completely separated in Min+(G) since Min+(G) is normal. Consequently, A, Bare completely separated in Min+ ( G) and therefore P is c embedded in Min+ ( G) by the Urysohn Extension Theorem. Finally, by [GJ, 6.5], the closed subset of Min+(G) of minimal prime subgroups in Q contains an homeomorphic copy of /JN. Let Ube a free ultrafilter on N. For g E G, let N(g) = {n: g E Pn}Define a new prime subgroup P by g E P if and only if N(g) EU. We show that Pis a
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29 minimal prime subgroup. The following proof is the same as that for [HJ, 4.8] and for [HLMW, 4.1). Let g, h E P. Pis a subgroup since N(g h) ::) N(g) n N(h) EU implies g h E P by filter properties. By convexity and since the Pn are prime subgroups, N(gV h)::) N(g) EU and N(gAh)::) N(g) EU, we have that Pis a sublattice of G. Thus Pis an subgroup of G. Let O g h E P. Then N(g) ::) N(h) EU since each Pn is convex, and thus Pis convex. Let gAh E P, then N(g)UN(h)::) N(gAh) EU implies that N(g) or N(h) is in U since U. Thus Pis a prime subgroup of G. Let g E P. Since Pn is a minimal prime subgroup for each n, we have that for each n E N(g), there exists an kn E G \ Pn such that g A kn= 0. By Lemma 2.3.2, there exists h E G such that hJ. = nneN(g) h;. Then g A h = 0 and h rJ_ Pn for all n E N(g) since hJ. h; Pn for each n E N(g). Thus N(g) n H(g) rJ_ U and hence hr/. P and Pis a minimal prime subgroup of G. Recall that a space X is cozerocomplemented if for any cozeroset U X there exists a cozeroset V X such that U n V = 0 and U U V is dense in X. A concrete example of a maximal ideal of infinite rank is found in C(X) where X is cozerocomplemented and first countable. DeMarco shows in [Le, ] that rk(Mp) 2'.: 2, for any nonisolated point of a first countable space (the result actually says more than this, and we will discuss this in Chapter 4). By modifying DeMarco's proof, we show that Mp contains infinitely many minimal prime ideals and hence has rank at least 2'. Proposition 2.3.4. Let X be first countable and let p E X be nonisolated. For every m E N there exists a family of m distinct prime ideals which sum to Mp. Moreover, if X is also cozerocomplemented, then there exist at least 2' minimal prime ideals contained in Mp.
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30 Proof: Since p has a countable base, there exists g E C(X)+ such that Z(g) = {p}. Let {}~ 1 be a neighborhood base at p and define a sequence of real numbers { 1 be given and let U 1 ,U 2 ... ,Um be distinct free ultrafilters on the sequence {xn}Define P; = {f E C(X): 3A E U,A Z(f)}, for each i. DeMarco shows that each of these sets is a prime zideal of C(X) and that MP is the sum of any two. Thus these prime ideals are noncomparable. We will demonstrate, as DeMarco has done for m = 2, that Mp is the sum of these m noncomparable prime ideals. Let {Ai}~ 1 be a collection of m pairwise disjoint subsets of {xn} such that A; EU, for each 1 i m. If Bi = g(Ai), then Bi U {0} is a closed subset of JR. Thus Bi U { 0} is a zeroset of JR. Choose 'Pi E C (R.) for each 1 i m such that Z(cpi) = Bi U {0} and E~ 1 'Pi= lJR. Let ui = 'PiY Then Ai C Z(g) = Z(ui), hence ui E Pi. Finally, we have g = u1 + u2 ++Um. If h E Mp, then Z(h+ g) = {p} and Z(h+ + g) = {p} By the above, h+ + g, h+ g E E~ 1 P; and hence h+, h, h E E~ 1 P;. The final statement of the proposition follow from the previous proposition, since the cardinality of /JN is 2', by (GJ, 9.2]. Now that we have illustrated the extreme cases of 0,1 and infinite rank, we present a result of (HLMW, 3.1], which gives a test for finite rank of a point of a compact space.
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31 Proposition 2.3.5. Let X be a compact space. Then p E X has rank n < oo if and only if there exist n pairwise disjoint cozerosets {U;}j=l with p E n;=l cl(U;), and no larger family of pairwise disjoint cozerosets has this property. An /ring A is called an SVring if A/Pis a valuation domain (i.e. principal ideals are totallyordered) for every prime ideal P. A space X is an SVspace if C(X) is an SYring. We will discuss this class of rings in more detail in Chapter 3. However, using the above, it is shown in [HLMW, 4.1] that any compact SYspace has finite rank. The validity of the converse of this result is unknown. Presently, our objective is to show that the result in Proposition 2.3.5 does not hold for infinite rank. To demonstrate this, we define a cardinal function on compact spaces and compare its value with a known cardinal invariant. Definition 2.3.6. Let K be a cardinal and let X be a compact space with p EX. Let {Ua}a<~ be a family of pairwise disjoint cozerosets of X and call the set na<~ cl(Ua) a Kboundary. Define p(p, X) to be the infimu.m over all {infinite) cardinals K such that p is not contained in a Kbou.ndary and let p(X) be the supremum over all the points p E X of the cardinals p(p, X). Recall that the cellularity of a space X, denoted c(X) is the infimum over all (infinite) cardinals K such that every family of pairwise disjoint open sets of X contains at most K many sets. Proposition 2.3.7. Let X be compact Then p(X) c(X)+. Proof: If p(X) > c(X)+, then there exists a x;boundary of cardinality greater than the cellularity of the space, which is nonsense. Example 2.3.8. Let > No. The product space, 2r, where 2 is the twopoint discrete space is called the Cantor space of weight ,. We show in Example 4.3.4
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32 that every point of 2r has infinite rank, thus rk(C(2r)) 2' by Proposition 2.3.3. By [E, 3.12.12(a)], we have that c+(2r) = N 1 Thus, Proposition 2.3.7 gives us that N 0 < p(2r) N 1 Therefore, since N 1 =I= 2', we see that p(2r) i= rk(C(2r)). D 2.4 Rank via z#Irreducible Surjections We must first recall a few definitions from (Ha] and (HVW2]. Let X, Y be Tychonoff Hausdorff spaces. Let f : Y X be a surjective continuous map. Then f is perfect if it is a closed map such that the inverse image of any point is compact. A perfect map is irreducible if proper closed sets of Y map to proper closed sets of X. The pair (Y, !) is a cover of X if/ is a perfect irreducible surjection from Y to X. Let (Yi, / 1 ) and (Y2, fi) be covers of X. We define (Yi, !1) ~ (1';, !2) if there exists a homeomorphism g : Y 1 Y 2 such that fig = / 1 Order the set of ~equivalence classes of covers via: (Y 1 Ji) (Yi, fi) if and only if there exists a continuous map g : Y 1 Yi such that fig= J 1 A class of spaces C is a covering class if for any space X there exists a least cover (Y, !) of X such that Y E C. The minimal extremally disconnected, basically disconnected and quasiF covers of compact spaces are described in (PW], (V], and, respectively, in (DHH], [HudP], and [HVWl]. Certain covering maps allow us to compute the rank of a space externally. A perfect irreducible surjection : Y X is z#irreducible if for each cozeroset U Y, there is a cozeroset V X such that cly(U) = cly(4>1 (V)) This condition on maps is also known as sequential irreducibility and w 1 irreducibility. It turns out that a map is z#irreducible if and only if C ( ) is a rigid embedding of C ( X) inside C(Y), by (HaM, 2.2]. Hence we have a homeomorphism Min(C(Y)) ~ Min(C(Y)) via contraction, by (CM, 2.3]. It is therefore not surprising that these maps are useful for calculating rank.
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33 Example 2.4.1. Let X be a compact space. The quasiF cover of X, (QFX, x), constructed in [HVWl] has the property that x is z#irreducible. We summarize this construction. Let z#(X) = { clxintx(Z) : Z E Z(X)}. For A E z#(X) denote the set of ultrafilters on z#(X) containing A by A. The authors of [HVWl] show that T(Z#(X)), the compact space of ultrafilters on z#(X) whose topology has a closed base given by { A : A E z# ( X)}, is quasiF and define a perfect irreducible surjection x: T(z#(X)) X by a E T(Z#(X)) maps to the unique point in n{A: A Ea}. Then QF X = T(Z#(X)) with the map x is the quasiF cover of X. The map x is z#irreducible: It is shown in [HVWl, 2.9] that if we have A E z#(QFX), then x(A) E z#(X), which is equivalent to the property of z#_ irreducibility. In fact, the quasiF cover of X is characterized up to equivalence in [HVWl, 2.13] as the only cover (Y, J) of X for which Y is quasiF and f is z#irreducible. Before we continue, we discuss the question (now answered) which was our motivation for considering this line of investigation. Let X be a compact space of finite rank and W = {x EX: rkx(x) > 1}. In [La2], Suzanne Larson asks if Wis always closed in X. The answer is no. Her counterexample, presented at ORD98 (a conference on lgroups held in Gainesville, FL in 1998) follows: Example 2.4.2. Let Ube a free ultrafilter on N. Let E =NU {a} where points of N are isolated and neighborhoods of a are of the form U U {a}, where U E U. Let E; = E for j = 1, 2 and define Y = (E 1 II E 2 )/(a 1 ~ o2 ). Let Y,. = Y for each r E R, and let X = (llrelR Y,.) U { oo} where neighborhoods of oo contain all but countably many copies of the Y,.. Then oo is a Ppoint which is in the closure of the set W = {x EX: rkx(x) > l}.
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34 We provide a characterization of points of finite rank of a compact space X via z#irreducible maps onto X. Let B be an fsubring of A and let O denote the natural surjection Max(A) Max(B). The following is proved in [HaM, 2.5]. Lemma 2.4.3. Let A and B be commutative !rings with identity and bounded inversion. Let B be an fsubring of A. Then if B is rigid in A, we have that O(OM) = n{O(N) n B : ON= OM}, and if ONi = OM (for j=l,2} with Ni II N2, then (O(N1) n B) II (O(N2) n B). Let X and Y be compact spaces and : Y X a z#irreducible map. Then 0: Max(C(Y)) Max(C(X)) is given by M .+ {f E C(X) : f EM}. Lemma 2.4.4. Let X and Y be compact and : Y X a z#irreducible map. Let p E X then O(Mq) = Mp if and only if (q) = p. Therefore, we have that Op= n{Oq n C(X) : q E <1>1 {p} }. Proof: Let q E <1>1 {p}. If f E O(Mq) then f E Mq and therefore we have that 0 = f(q) = f(p) and f E Mp. Let g E Mp; then (g)(q) = g(p) = 0, and hence g E O(Mq)Thus O(Mq) = Mp. Conversely, assume (q) = r f. p. By complete regularity, there exists f E C(X) such that f(p) = 0 and f(r) = 1. We have / E Mp, but f(q) = J(r) = 1 f. 0 and hence f O(Mq)The final statement then follows from the first and Lemma 2.4.3. Proposition 2.4.5. Let X and Y be compact spaces and : Y X z#irreducible. 1. If p EX such that rkx(p) = n, then l1 {p}I = n. 2. If Y is an Fspace and l<1>1 {p}j = n then rkx(p) = n. S. If Y has finite rank and l1 {p}j = n, then rkx(p) < oo. Explicitly, we have that rkx(p) = E~ 1 rky(qi) where <1>1 {p} = {qi}f=i
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35 4. Ifp EX is an Fpoint of X, then q E 4>1 {p} is an Fpoint of Y. Proof: (1) Let 4>1 {p} = {qi}iEI for some index set J. For each i E J, choose Qi E Min(C(Y)) such that Oqi Qi. Then by Lemma 2.4.4, it follows that we have Op= niEr(Oqi n C(X)) Qin C(X) Mp, for every i E J. By the bijection described in Proposition 2.2.7, the set of minimal prime ideals contained in Mp is given by { Qi n C(X)}iEI and III = n. ( 2) Let 4>1 {p} = {Qi} i=l. If Y is an Fspace, then O qi is prime for each i and we have that Op= nf= 1 (Oqi n C(X)) Oqi n C(X) Mp for each i. Thus, by the bijection described in Proposition 2.2.7, the subspace of minimal prime ideals contained in Mp is {Oqi n C(X)}f= 1 and rkx(p) = n. (3) Let 4>1 {p} = {qi}f= 1 and let the subspace of minimal prime ideals in Mq be given by {Qi;: 1 j rky(qi)}i=l Then as above, the subspace of minimal prime ideals contained in Mp is given by {Qi; n C(X) : 1 j rky(qi)}i=: 1 and hence we see that rkx(p) = E:= 1 rky(qi) < oo, as desired. (4) Let p EX be an Fpoint and 4>1 {p} = {q}. Then Op= Oq n C(X) Mp is a minimal prime. There exists a unique Q E Min(C(Y)) such that OP= Q n C(X). Since Oq n C(X) = Q n C(X) and Q is unique, Oq = Q E Min(C(Y)). If Y = QF X in Proposition 2.4.5, then the third statement is a partial converse of [HLMW, 5.1] which states that if Xis compact and has finite rank then QF X has finite rank. The final statement is an extension of [HVWl, 3.12] in which it is shown that the preimage of a Ppoint of Xis a Ppoint of QFX. The second statement says that if QF X is an Fspace, then the points of X of rank one are precisely the points with unique preimage under the covering map x. In this light, one should ask when a quasiF cover of a space is an Fspace. Recall that a space X is fraction dense if the classical ring of quotients of C(X) is rigid in the maximal (Utumi) ring of quotients of C(X). In [HVWl, 2.16],
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36 it is demonstrated that Q F X is basically disconnected if and only if X is cozero complemented. In fact, the basically disconnected cover is the quasiF cover in this case, see [HaM, 2.6.2). The fact that QFX is realized by the extremally disconnected cover if and only if Xis fraction dense is proved in [HaM, 2.4). By [HudP, 6.2), QFX is an Fspace if and only if for any two disjoint cozerosets Ci, C2 X, there exist Z 1 Z 2 E Z(X) such that Ci Zi for i=l,2 and int(Z1 n Z2) = 0. We now provide an example of a space X such that QF Xis an Fspace which is not basically disconnected. Recall that a space is acompact if it is a countable union of compact spaces. Lemma 2.4.6. Let X be a noncompact acompact locally compact Fspace which is not basically disconnected. Let X 1 = X 2 = {3X and define Y be the quotient space of the topological sum of X1 and X2 where pairs of corresponding points of X; \ X (j=1,2} are collapsed to a single point. Then Y is not quasiF and QFY is an Fspace which is not basically disconnected. Proof: The disjoint union U = X II X is a dense cozero set of Y which is not C*embedded in Y. Thus Y is not quasiF. The quasiF cover of Y is X1 II X2, which is an Fspace but not basically disconnected. Example 2.4. 7. Let X be the disjoint union of a countable number of copies of the corona {3N \ N. Then Xis a acompact noncompact Fspace which is not basically disconnected. Construct Y as defined in Lemma 2.4.6. Then QFY is an Fspace which is not basically disconnected. 2.5 Prime Character The second character we consider counts the minimum number of minimal prime subgroups that we must sum in order to obtain the lex kernel of an group.
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37 Definition 2.5.1. Let S be a family of minimal prime subgroups of G. We call S ample if VS= lex(G). The prime character of G, denoted 1r(G), is the least cardinal so that any family of distinct minimal prime subgroups of that cardinality is ample. Note that iflex(G) = 0, i.e., if G is a totallyordered group, then we say 1r(G) = l. A prime subgroup properly contained in lex( G) is called embedded. Proposition 2.5.2. Let G be an lgroup and m a positive integer. The following are equivalent: 1. 1r(G) = m < oo. 2. m = 1 + sup{rk(P) : PE Spec+(G) is embedded}. S. m is minimal with respect to the property that lex(G) = G(LJ; 1 af) for any m pairwise disjoint positive elements, {a;}f= 1 lex(G)+. 4m is minimal such that for any embedded prime P, the chains of proper polars in P have length at most m 1. Proof: (1) =} (2) : Let P be embedded If P contains the m minimal prime subgroups {Q;}f= 1 Min+(G) then Vf= 1 Q; P lex(G). Hence 1r(G) > m. Thus we have shown that 1r(G) m implies rk(P) < m. Thus by (1), rk(P) m 1 and sup{rk(P) : P embedded} m 1. If sup{rk(P) : P embedded} < m 1, then for any family S of m 1 minimal prime subgroups of G, VS is not embedded since rk(VS) m1. Thus VS= lex(G) and 1r(G) m1. Thus 1r(G) = m implies that sup{rk(P) : P embedded} m 1. Therefore sup{rk(P) : P embedded} = m + 1. (2) =} (3): Let {a;}f= 1 lex(G)+ be pairwise disjoint. Let g E lex(G) \ G(LJ; 1 af) and let V be a value of g such that G(LJ; 1 af) C V C lex(G). Then by the polar characterization of O(V), [BKW, 3.4.12}, we have that a; (j_ O(V) for each j such that 1 j m. By (2), V contains at most m 1 minimal prime subgroups, {Qi}~ 1 1 Since O(V) = n~~ 1 Qi, we have by the pigeonhole principle that there
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38 must be a minimal prime subgroup, Q, contained in V which fails to contain two of the elements of {a;}j=i But since these elements are pairwise disjoint, this contra dicts the fact that Q is prime. Therefore, lex(G) = G(Uf: 1 a,f ). For the minimality of m, let P be an embedded prime of rank n < m. Then by the Finite Basis Theorem, P/O(P) contains n pairwise disjoint elements of corresponding to elements of P which are not in O(P), say {bAJ:= 1 s; P. Then bf C P for each k such that 1 :5 k $ n and G(UZ=i bt) s; P lex(G). (3) :::} (1) : Assume that 1r(G) > m. Then there exists S = {Q;}j= 1 C Min+(G) which contains m elements and is not ample. Let Q = VS lex( G). For each j, let q,; E Qt\ Q;. Then q, = Vfa:= 1 q,; E Qt\ (UA:;ei QA:)Disjointify by defining q, = A,;e;Q.; Af= 1 Qk r/. Q,. Then Q; E Q, for every j =/: i and then we obtain that G(U;: 1 q/) s; VS~ lex(G). Hence, 1r(G) $ m. By the minimality of min (3) and by (1):::} (3), we must have 1r(G) = m. (3) # ( 4) : Follows directly from the Finite Basis Theorem [D, 46.12] applied to P/O(P) for any embedded prime P. The following is immediate: Corollary 2.5.3. Let G be an lgroup. The following are equivalent: 1. 1r(G) < oo. 2. sup{rk(P): P embedded}< oo. 3. There exists m E N such that lex(G) = G(lf= 1 af) for all families of m pairwise disjoint elements {a;}f=i We now consider some grouptheoretic properties of the prime character. Note that for groups A and B, AEBB denotes the group Ax B with componentwise operations and is called the cardinal sum.
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39 Proposition 2.5.4. Let G be an group. 1. For any homomorphic image, H, we have 1r(H) :5 1r(G). 2. For any CE
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40 character. I '(2.1) Let r be the root system above, where the subgraphs r A and r B each have infinitely many identical branches descending from its maximal vertex. Define G = E(r, JR), A= {v E G: supp(v) c r A} and B = {v E G: supp(v) c rB}Then we have that A~ E(r A,1R), B ~ E(rB,1R) and AV B ~ E(r Au fn,lR). Now, it is evident that 1r(G) = oo, 1r(A) = 3 = 1r(B), and 1r(A VB) = oo. D Recall that an group G has a finite basis if it contains a finite maximal set of elements {b; }j= 1 such that the set {g E a+ : g b;} is totally ordered for each j. The following indicates when we can expect 1r(A VB) < oo: Proposition 2.5.6. Let G be an group and let A, B be convex subgroups such that 1r(A) = m < oo and 1r(B) = n < oo. If lex(A VB) = lex(A) V lex(B), then 1r(A V B) < oo. Otherwise, 1r(A V B) < oo if and only if each of A and B has a finite basis. Proof: Assume that lex(A VB) = lex(A) Vlex(B) and let P lex(AV B) be a prime subgroup of G. Then either lex(A) %. P or lex(B) %. P, or both. Say lex(A) %. P. Then P n lex(A) is an embedded prime subgroup of A, and since 1r(A) = m, we have that rkA(P n lex(A)) < m 1. Then by [BKW, 2.4.7], rkAvB(P) < m 1. Likewise, if lex(B) %. P then rkAvB(P) n 1. Thus, for every prime subgroup P lex(A VB), we have that rkAvB(P) max{m1,n 1} < oo. Therefore, by Proposition 2.5.2, 1r(A VB) < oo.
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41 Note that we always have that lex(A) V lex(B) lex(A VB). We assume now that lex(A VB) =/= lex(A) V lex(B) and let PE Spec+(A VB) have the property that lex(A) V lex(B) P lex(A V B). If each of A and B has a finite basis, then rkAvn(P) = IMin+(A)I + IMin+(B)I < oo and for any embedded prime subgroup Q of AV B, we have that rkAvn(Q) ::; IMin+(A)l+IMin+(B)IThus, by Corollary 2.5.3, we have that 1r(Av B) < oo. Conversely, 1r(AV B) < oo implies that rkAvn(P) < oo. Hence, since IMin+(A)I, IMin+(B)I < rkAvn(P), we have that each of A and B has a finite basis by the Finite Basis Theorem. The proof of the following is evident: Proposition 2.5.7. Let G be an group and let A, BE v(,) = O}. If v He,
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42 then there exists E C such that v(,) :/= 0. Since supp(v) is finite, v is in all but finitely many minimal prime subgroups of E(r, JR). Thus by Proposition 2.5.8, E(r, R.) has Property F. Proposition 2.5.10. If G is a finitevalued group of finite prime character, then lex(G) has Property F. Proof: Let g E lex(G). Then any minimal prime subgroup oflex(G) not containing g is contained in a value of g. Since each value of g contained in lex(G) contains a finite number of minimal prime subgroups and there are only finitely many values of g, there are only finitely many minimal prime subgroups of lex(G) not containing g. Thus lex( G) has Property F. Proposition 2.5.11. Let G be an group and let m be a positive integer. If m is minimal such that every pairwise disjoint subset of G contains at most m 1 elements, then 1r( G) = m. Proof: Any proper prime subgroup of G is contained in a value of G, hence proper prime subgroups of G contain at most m 1 minimal prime subgroups. Thus 1r(G) :'.5 m. Let n < m and assume that every family of n minimal prime subgroups of G is ample. Then every proper prime subgroup (in particular, every value) of G contains at most n 1 minimal prime subgroups. This contradicts the minimality of m. Thus 1r( G) = m. Example 2.5.12. The following is an example of an group of finite prime char acter such that lex( G) has Property F but G has pairwise disjoint sets of any size m. Thus the converse of Proposition 2.5.11 does not hold. Let r be the root system: I\ I\ I\ (2.2)
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43 and let G = E(r,R). Then 1r(G) = 3 and G is finitevalued. Thus, by Proposi tion 2.5.10, lex(G) has Property F. We demonstrate that there are bounded disjoint families of any given size. Index the maximal elements of r by;, where j is a positive integer, and let v; E G be such that v;(;) = 1 and supp(v;) c {'YE r: 'Y ~;}where v;(6) = 0 for all 6 < ;. Then { v;};~ 1 is an infinite pairwise disjoint family in G. Let a positive integer m be given. Choose any m elements from this set, { v; 1 vh, ... v;,..} and let s = uzi=l supp(v;.). Let V be the characteristic function on the finite set s, then v E E(r,R) and v v;. fork= 1,2, ... m. D Example 2.5.13. The converse of Proposition 2.5.10 does not hold. That is, we present an example of a finitevalued group with Property F and infinite prime character. Consider the following root system r which is indexed by the positive integers: I /\ Ii\ (2.3) Then each prime subgroup of the group G = E(r, R) contains a finite number of minimal prime subgroups, yet there is no bound on the number of minimal prime subgroups in each prime subgroup. Thus 1r( G) = oo. G has Property F by Corollary 2.5.9. D 2.6 Filet Character The third character that we define measures the length of a chain of incidents of branching. Definition 2.6.1. Let G be an group. C = {11,Q; E Spec+(G): i O,j 1} is called a filet chain of prime subgroups if Po 2 P1 2 P2 ... for all i, Ps II Qi, and
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44 ~+i V Qi+l ~~for all i 2:: 0 {see below}. (2.4) The length of a filet chain is given by l(C) = max{j E N : 3Q; E C}. If the maximum does not exist, we write l(C) = oo. The filet character (G) is given by ( G) = sup{ l ( C) : C is a filet chain}. If Spec+ ( G) has no filet chains, i.e., if G is semiprojectable, then we say that ( G) = 0. Proposition 2.6.2. Let G be an group. Then (G) $ 1 if and only if 1r(G) $ 2. Proof: Suppose that 1r( G) $ 2. If ( G) > 1 then there exists a filet chain C of length 2 in which we may assume that P 0 = lex(G). Since rk(P 1 ) = 2, we have that 1r(G) > 2, by Proposition 2.5.2. Conversely, assume that 1r(G) > 2. Then there exist minimal prime subgroups P 2 Q 2 such that P 1 = P 2 V Q2 =f:. lex(G). Thus for any minimal prime subgroup Q 1 % P 1 the set C = {P 0 ,P 1 ,P 2 ,Qi,Q 2 }, where P 0 = lex(G), is a filet chain of prime subgroups in G oflength 2. Therefore (G) > 1. For larger filet character, the relationship between it and the prime charac teris more complicated, as the following example illustrates. Example 2.6.3. Let r be the following root system: (2.5)
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45 Then G = I;(r, R.) has (G) = 2 while 1r(G) = 4. The following relationships between the characters hold: Proposition 2.6.4. Let G be an group. Then 1. (G) :5 rk(G) 1. 2. (G) :5 1r(G) 1 :5 rk(G). Proof: Let rk(G) = m. If G has a filet chain C = {Pi,Q; E Spec+(G): i 0,j 1}, then rk(Po) :5 m and hence l(C) :5 m1. Therefore, (G) :5 m1 = rk(G)1. If 1r(G) = n and C = {Pi, Q 3 E Spec+(G) : i O,j 1} is a filet chain in G then l(C) < oo since rk(P 1 ) :5 n 1 by Proposition 2.5.2. Hence, in fact, l(C) :5 n 1 and therefore, (G) :5 n 1 = 1r(G) 1. Now, 1r(G) :5 rk(G) + 1 by Proposition 2.5.2. Thus, finally, (G) :5 1r(G) 1 :5 rk(G). At this time, any statement that we make about the filet character of an group requires a restriction on the rank and prime character. Rather, we can not say much more than what we establish earlier in this chapter. We leave this investigation for a later date.
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CHAPTER 3 GENERALIZED SEMIGROUP RINGS Let A be a root system. Starting from the standard semigroup ring and Hahn group constructions and the investigation of the following section, we build an /ring 'R having A as the skeleton of the graph of Spect('R). We follow up on ideas presented in two papers of Conrad [Cl],[C2], in the paper of Conrad and Dauns [CD], and in the paper of Conrad and McCarthy [CMc]. The first two papers look at the conditions on A which will yield a ring structure on V(A, JR) and on the subgroup F(A, R.) of elements v for which supp(v) is the join of finitely many inversely well ordered sets in ti. The paper [CD] focuses on the case when V(fi, JR) is a division ring, while in [CMc] the conditions are established for the ring to be an ring its properties are studied when f1 is finite. Note that F(fi, :R) is denoted by W(fi, JR) in [Cl],[C2], and [CD]. 3.1 Specially Multiplicative {Rings Let r be a partially ordered group which is a root system. Suppose also that r is torsion free, that the subgroup H of r generated by the positive cone is totally ordered and that r / H is finite. In the paper of Conrad and Dauns [CD, 2.2], it is shown that V(r, JR) = F(r, JR) and that V(r, JR) is a latticeordered division ring under the usual groupring multiplication: for u, v E V(r, JR), and, Er u v(,) = I: u(a)v(,B). a+/J='Y It is easy to see that if an element of (V (r, JR), +, *) is special (i.e., has only one maximal component), then its multiplicative inverse is also special. Hence, the 46
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47 special elements of the ring V(r, R.) form a multiplicative group. Moreover, the following holds in general, [CD, Theorem I]: Theorem 3.1.1. Let R be a latticeordered division ring with identity. The follow ing are equivalent: 1. The special elements of R form a multiplicative group or the empty set. 2. If a E R+ is special, then a1 > 0. 3. If a E R is special, then a1 is special. 4. For all a ER+ special and x, y ER, a(x Vy)= ax Vay. Since the authors of the paper [CD] seek an embedding theorem for ifields, their investigation in this realm is restricted to division rings. In this section, we consider a class of /rings (which are not division rings) in which the special values form a partially ordered semigroup. The pursuit of a characterization similar to Theorem 3.1.1 of irings satisfying this condition is left for another time. For the/ rings that concern us here, some particular assumptions are needed on the associated semigroups. As a formality, we define: Definition 3.1.2. Call an fring A specially multiplicative if the special values of A, with an appropriately adjoined 0, form a partially ordered semigroup. Recall that an /ring A is called an SVring if A/Pis a valuation domain (i.e., the set of principal ideals is totally ordered) for every prime ideal P. Let g be the class of commutative /rings which are local, bounded (that is, A*= A), semiprime, finitevalued, finite rank and square root closed SVrings with identity and bounded inversion. We demonstrate that the elements of g are specially multiplicative and then investigate the properties of the associated semigroups. We must first remind the reader of a couple of facts about special values and of the relatively deep theorem, recorded as [HLMW, 2.14]:
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48 Theorem 3.1.3. Let A be an !ring of finite rank with identity and bounded inver sion which is local, semiprime and square root closed. Then A is an SV ring if and only if whenever O a b and b is special, there exists x E A such that a = xb. The proof of the following lemma is wellknown and routine, however, it is instructive, so we include it here. We remind the reader that we denote the root system of values of an group G by r( G) = {Vr : 'Y E f}, where r is a partially ordered index set that is orderisomorphic to f(G). Lemma 3.1.4. Let G be an group and let a, b E a+ be distinct special elements with values at V 0 V.8, respectively. 1. Vo II Vp if and only if a A b = 0. 2. V 0 < Vp if and only if a b. Proof: First, assume that Va II Vp. We show that a Ab is contained in each value of G and conclude that a and b are disjoint. Let V be a value in G and assume that b V. Then V Vp and a E V since, else, V Va, which contradicts the assumption of incomparability. Thus, by convexity we obtain that a A b E V. Again by convexity, if b E V then a Ab E V. Thus a Ab = 0, as desired. Conversely, assume that a and b are disjoint and, by way of contradiction, assume ( without loss of generality) that Va~ V.8 Since a V 0 and V 0 is a prime subgroup, b E V 0 Vp, a contradiction. Second, assume that Va< Vp. Then a E V.8 If there exists an integer n such that na > b, then since b V.8, we must have that na Vp, by convexity. But this contradicts that a E V.8, and so we conclude that a b. Conversely, assume that a~ b. We know that b V 0 and hence Va Vp. If Va = Vp, then by [D, 12.6], there exists an integer n such that na > b, which is nonsense. Thus V 0 < V.8.
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49 Combining Theorem 3.1.3 and Lemma 3.1.4 leads to the main result of this section: Theorem 3.1.5. Let A E <;;. 1. If a, b E A + are special and not disjoint, then ab is special 2. If a, a' are special with value Va, b, b' are special with value Vp and Va, V p are comparable then the special value of ab is the same as that of a'll. Proof: Let a, b E A+ be special with values at Va, Vp, respectively, and let M be the maximal ideal in A. Since A is bounded, we may assume without loss of generality that Va, Vp < M. This gives rise to the relations a, b 1 and ab a, by Lemma 3.1.4(2) Theorem 10.15 of [AF] states that an abelian group is finitevalued if and only if each positive element is a finite disjoint sum of positive special elements. Thus we assume, by way of contradiction, that ab= c1 + C2 ++ck, where each c; E A+ is special and c; /\ Cj = 0 for all i =I= j. Then O < c; ab < a for each i and therefore Theorem 3.1.3 gives the existence of Xi E A + such that c; = ax i for each i = 1, ... k. Without loss of generality, we may assume that O < Xi b, by replacement with xi /\ b. Assume that xi = E~i Xi ; is a decomposition of X i into a sum of special elements. Then c; = E;~i axi; implies that axi ; = 0 for all but one j, by [AF, 10.15 ] Thus we may choose each xi to be special, without loss of generality. For each i, let Vo be the value of XiThen b Vo. and therefore for all i we have that Vo Vp. If Vo < Vp, then xi b. This gives a contradiction, since it implies that ax i ab. On the other hand, if Vo, = Vp, then there exist n, m E N such that x i ~ nb and b mx i Therefore, axi nab and ab~ maxi Thus by [D, 12.6] we have that the value of c; is the unique value of ab and ab= axi. Hence, ab is special.
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50 For the second statement, let be the value of the special element ab. We show that is the value of all and then a similar argument and transitivity gives that Vo is also the value of a'b', as desired. Towards this end, let V 7 be the value of ab'. H V 7 II Yo, then O = ab A all= a(b Ab') and hence O = a A (b All). On the other hand, if Va Vp, then since b All V 0 we have that a E Va, H Vp Va, then since a Vp, we have b Ab' and hence b or ll is in V13. These contradictions lead us to conclude that V 7 or V 7 V 0 If Vo < Vy then ab~ ab'. However, by [D, 12.6], there exists an integer n such that nb > ll and therefore, nab> nab'. Likewise, V 7 1'l,'o. Thus V 7 =,as desired. Let A E g, Then by [AF, 10.10], all the elements of r(A) are special since A is finitevalued. Abusing notation, we now identify values with their indices and define an operation on r. Let a, b E A be special elements with corresponding values at a, /3, respectively. Append a generic symbol, such that II a for all a E r and define a multiplication on r U {} such that a = = a and a. /3 = { if a II /3, the value of ab otherwise. By Theorem 3.1.5, this operation is welldefined. Some properties of this operation are recorded in the following proposition. Define a ~ /3 if and only if a and /3 are contained in the same maximal chains of r. Then~ is an equivalence relation on r. Let r O denote the ~equivalence class of a. Proposition 3.1.6. Let A E g and let a, /3, ,, 8 E r correspond to the values of the special elements a, b, c, d E A+, respectively. Then 1. The operotion is associative, for every a, /3, 1 : ( a /3) 1 = a (/3 1 ) 2. If M is the maximal ideal of A, then M is a value of the identity. Let = M. Then a = a = a for every a
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S. If a {3 ::/=
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52 d = yb. Then O = ab I\ yb = ( a I\ y )b gives that O = a I\ y I\ b and hence O = ayb = ad and therefore al\d = 0. This says that a II 6, a contradiction. We now have that a /3 is comparable with any 6 < {3. It follows from this fact and from (3) that maximal chains are closed under the operation. (7) Let,, 6 Er 0 By (6), we have that, 6 is in the same maximal chains as, and 6. Hence 1 6 Er 0 (8) We know that, < a if and only if Vy < V 0 if and only if O < c a. This implies that there exists a positive element b such that c = ab, by Theorem 3.1.3. We have, by (AF, 10.15), that b is special. Thus, if V.B is the value of b, then = a {3, as desired. Corollary 3.1. 7. Let A E Q. Then A is specially multiplicative. Moreover the operation is a surjection onto r (A) U { }. 3.2 rSystems and Systems Let Ci be a given root system. The discussion of the previous section and of the papers (Cl], (C2], (CD], and (CMc] prepare us to construct an !ring having Ci orderisomorphic to its root system of values. However, under the assumptions placed on Ci in these papers, these rings sometimes are rings, and rarely are !rings; in fact, as mentioned before, the paper (CD] focuses on the case when V(fi, JR) is a division ring. Also, contrary to our intentions, (CMc] concentrates on the properties of the ring when the root system is finite. We modify all these conditions on the root system in order to get a ring multiplication, *, yielding an fring structure on R = (F(fi,JR),+,*) such that the subring = (I;(~,JR),+,*), as defined in Chapter 2, is an fsubring. We start with (Cl,Theorem I] which establishes the ring structure. Proposition 3.2.1. Assume that Ci is endowed with a surjective partial binary op eration + : A Ci defined on A Ci x Ci. Let u, v E V ( Ci, JR), and define for
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53 8 E !l, u *v(8) = :Ea+.B=o u(a)v(.B). Then both~ and R are closed under* In fact, and 'R, are rings if and only if the operation on !l is (BaerConrad) associative: (a, /3), (a+ /3, 'Y) EA if and only if (/3, 'Y), (a, /3 +'Y) E A, and if (a, /3), (a+ /3, 'Y) E A then ( a + /3) + 'Y = a + (/3 + 'Y). Certain properties of these rings, and R, are completely determined by the operation + on /:l; commutativity is one such attribute. Definition 3.2.2. Let (!l, +) be a root system with a partial binary operation, +. We call (!l, +) an rsystem if the operation is surjective and (BaerConrad) asso ciative. An rsystem (!l, +) is commutative if (a, /3) EA if and only if (/3, a) E A and a + /3 = /3 + a for all such pairs. Proposition 3.2.3. Let (!l, +) be an rsystem. (!l, +) is commutative if and only if R is a commutative ring. Proof: If (!l, +) is commutative, then for u, v E 7l we have u v(8) = L u(a)v(/3) = L v(f3)u(a) = L v(f3)u(a) = v u(8). a+,8=6 a+,8=6 .B+a=6 Thus 'R, is a commutative ring. Conversely, if a+ /3 =/= /3 + a, then Xa+.B = Xa X.B =I= X.B Xa = X.B+a In order to obtain an ring, we must ask that the operation on /:l preserve the order of the root system and restrict the domain of the operation. AB we will indicate, the various strengths of order preservation and restrictions of domain yield varying richness of order structure on the rings. Definition 3.2.4. Let /:l be a root system. An rsystem (!l, +) is an system if it satisfies: a < /3 and (a, 'Y) E A implies that (/3, 'Y) E A and, in this case, a+'Y < /3+'Y; and if ( 'Y, a) E A implies that ( 'Y, /3) E A and, in this case, 'Y+a < 'Y+ f3. If every connected component of /:l has a maximal element, we call /:l bounded above.
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54 Note that the system condition gives that no nonmaximal element of such a bounded above root system is idempotent. The following theorem is in (CMc, ). Theorem 3.2.5. Let (Ll, +) be an rsystem. Then 'R, is an iring if and only if Ll is an system. In the next section, we describe the conditions on an system which induce an fring structure on 'R,. As mentioned, there are some restrictions that we need to place on the domain and range of the operation. Before we discuss this situation, we present an example of an system that gives rise to an ring which is not an fring. Probably, this is the simplest example of such an system. Example 3.2.6. Let ,X > a > 2a > 3a. . and > /J > 2/3 > 3/3 .... Then we let Ll 0 = {.X} U {na}~= 1 let Llp = {} U {n/J}~=l and totallyorder Ll 0 x Llp lexicographically such that (8 1 , 1 ) < (82, 2 ) if and only if 81 < 82 or we have 81 = 82 and 1 < 2 Identify pairs with sums and let Lla+/3 = {8+,: (8,,) E Ll 0 x Llp}. We define an associative and commutative addition on the root system given by disjoint union tl = tl 0 II tlp II Lla+/3 as shown in the following table, where v = A + and k, l, m, n are positive integers (note that the table is completed by reflection across the diagonal): + a /3 A V A+m/3 ka+ na +l/3 a 2a a+/3 a a+ a+ a+m/3 lk +lla+ ln+ lla +l/3 /3 2/3 A +/3 /3 A +/3 A+(m+l)/3 ka+/3 na + (l + 1)/3 A A V V A+m/3 ka+ na + l/3 V A+m/3 ka+ na + l/3 V V A+m/3 ka+ na + l/3 A+m/3 A+2m/3 ka+m/3 na+lm+fl/3 ka+ 2ka+ (k+n)a+l/3 na + l/3 2na + 2l/3 Let u, v E 'R,+. We need to show that u v E 'R,+ in order to conclude that 'R, is an ring. Since u and v are positive, it is evident that U*v(.X), U*v(), U*v(v) 0. If 0 = u v(.X) = u(X)v(.X), then assume that u(.X) = 0. For an integer n, u v(na) = u(X)v(na) + u(na)v(.X) + L u(ia)v(ja). i+j=n i,#0
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55 H u(na) = 0 for all n, then u v(na) = 0. Conversely, if n # 0 is minimal such that na E supp(u), then u v(na) = u(na)v(A) 0. Thus, if v(A) > 0 then na is the maximal element of supp( u v) below A. ff v(A) = 0 and v(ma) = 0 for all m, then u v vanishes on ..6. 0 Otherwise, let m be minimal such that ma E supp ( v). Let k = n + m then ka is maximal in supp(u v) and u v(ka) = u(na)v(ma) > 0. A similar computation demonstrates that u v > 0 at the maximal element of its support in AfJ also. Assume that O = u v ( v) and, when they exist let Au= max(supp(u) n ..6. 0 ) Av= max(supp(v) n Aa) JJu = max(supp(u) n AtJ) ,; = max(supp(v) n AtJ) <>u = max(supp(u) n Aa+fJ) <>v = max(supp(v) n Aa+fJ) (3.1) Then the maximal element of supp( u v) lying below v is given by and so we are left to show that u v(,) > 0. We have three cases to check here, namely 'Y = A +m/3, ka+ or na+l/3. Upon consideration of the formulae below, it quickly becomes clear that the only nonzero summands of u v ( 'Y) are those of the form u(a)v(r), where a and r are one of the six maximal support elements listed above in (3.1), which implies that the summands are all positive. For the sake of a certain degree of completeness of exposition, we list the possible summands in the case that 'Y =A+ m/3. The analysis in the other cases is similar. u v(,\ + m/3) = u(,\)v(m/3) + u(mf3)v(,\) + u(,\)v(,\ + m/3) + u(,\ + mf3)v(,\) + u()v(A + m/3) + u(,\ + mf3)v() + u(v)v(,\ + m/3) + u(,\ + mf3)v(v) + L u(,\ + if3)v(jf3) + L u(i/3)v(,\ + j/3) + L u(,\ + if3)v(,\ + j/3) i+j=m i+j=m i+j=m i,j#) i,#0 i,#0 To compute u v(,\ + m/3), consider all the possible combinations of the following situations and then add.
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56 1. H u(X) =/:0 then if v(m/3) =/:0, we have that m/3 v, If m/3 < v, then 'Y = ,\ + m/3
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u v(na + l/3) = u(na)v(lf3) + u(lf3)v(na) + u(A)v(na + lf3) + u(na + lf3)v(A) + u()v(na + l/3) + u(na + lf3)v() + u(v)v(na + l/3) + u(na + lf3)v(v) + u(na)v(A + l/3) + u(A + lf3)v(na) + u(na + )v(l/3) + u(lf3)v(na + + L u(ia + lf3)v(ja) + L u(ia)v(ja + l/3) + L u(na + if3)v(jf3) i+j=n i,#0 i+j=n iJf,O i+j=l iJ#O 57 + L u(if3)v(na + j/3) + L u(ia + )v(ja + l/3) + L u(ia + lf3)v(ja + i+j=l iJ#O i+j=n iJ#O i+j=n iJ#O + L u(A + if3)v(na + j/3) + L u(na + if3)v(A + j/3) i+j=l iJf,O i+i=l iJ#O + L u(ia + kf3)v(ja + m/3) i+j=n k+m=l iJ,k,m#O 'R, is not an fring: Let HllaH = {v E 'R,: 8 E /J.o.+fJ => v(8) = O} be the minimal prime subgroup. Then Xo., XfJ E Hlla+f:l' yet Xo. XfJ = Xo.+fJ Hlla+f:l Thus, Hlla+f:l is not an ideal and 'R, is not an !ring by [BKW, 9.1.2]. The following two propositions record some consequences of the lsystem condition. Note that none of the excluded conditions occur in Example 3.2.6. Proposition 3.2. 7. Let (JJ., +) be a bounded above lsystem in which maximal ele ments act as an additive identity on elements below it. Then the Jollowing can not occur for nonmaximal elements a, f3 E JJ., 1. /3 +a< a, where (3, a are below the same maximal element, and f3 II f3 + a 2. /3 < a /3 + a 4If A, E /J. are maximal and (A,) EA, then A+ {A,.
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58 In particular, the second and third properties imply that if a, {3, {3+a are comparable, then {3 + a < a, {3. Proof: If {3 + a < a, {3 II /3 + a, and > a, {3 is maximal, then XfJ, X, Xa are positive. Yet, XfJ (X, Xa) = XfJ XfJ+a is negative, contradicting that R is an ring. Assume that {3 < a {3 + a. Let {3 + a be maximal. Then as above, XfJ (X, Xa) < 0. The same contradiction is obtained in the case that we assume {3 {3 + a a < If)., are maximal and ). + < then X>. (X, 2X>.+,) = x>.+, < 0. Proposition 3.2.8. Let(~,+) be an system 1. There is no nonmaximal o E such that o is idempotent and o + a = o for all nonmaximal a > o. 2. If o E is nonmaximal and idempotent then there does not exist a maximal element > o. Proof: If o E such that o is idempotent and o + a = o for all nonmaximal a > o, Then we contradict the assumption that 'R, is an ring since Xo (xa 2xo) = xo If > o is maximal and o is idempotent, then Xo (X, 2xo) = Xo 3.3 (Systems We are much more interested in the /ring situation. In [CMc, 2.1 ), the authors demonstrate a condition on~ which will give rise to an /ring. Theorem 3.3.1. Let(~,+) be an system. Then 'R, is an /ring if and only i f the root system also satisfies: if a II /3 and (a,,) EA, then a+, II /3; and if ( 1 ,a) EA, then 'Y + a II {3.
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59 In this section we consider a condition on Li which is equivalent to the one stated above and proceed to investigate certain properties of the associated !rings. Recall that r(F(Li, JR)) is orderisomorphic to Li by Proposition 2.1.1, where the values are of the following form, for 8 E Li : = {u E 'R:, 8 => u(,) = O}. Definition 3.3.2. Let (Li,+) be an system such that 8 = a+ {3 implies a, {3 8. Then we say that (Li,+) is an J 0 system. Proposition 3.3.3. If (Li,+) is an J 0 system, then 'R is an ring and for every 8 E Li, the subgroup Vo is an ideal. Hence, in particular, 'R is an !ring and is an fsubring of 'R. Proof: Assume that (Li,+) is an J 0 system. Let 8 E Li, v E u E 'R, and 8. Let,= a+/3, then by the J 0 system assumption, {3 ~, 8 and hence v(/3) = 0, for all such {3 since v E Then U* v(,) = Eo+~=r u(a)v(/3) = 0 and we conclude that U*V E Therefore, the values of'R are ideals. Moreover, since each minimal prime subgroup is an intersection of a chain of values by [D,10.8], each is an intersection of a chain of ideals; thus each minimal prime subgroup itself is an ideal. Finally, we have shown that 'R, and~ are !rings by [BKW, 9.1.2]. Proposition 3.3.4. An system (Li,+) is an J 0 system if and only if 'R, is an !ring. Proof: Sufficiency is shown in Proposition 3.3.3. Conversely, assume that there exist a, {3 E Li such that a+ {3 ;:. a or a+ {3 ;:. {3. Then by Proposition 3.2.7, we have that a, {3, a+ {3 are not all comparable. If a II {3, then we have that a II a+ {3 by Theorem 3.3.1. Assume that {3 II a+ {3 and let C be a maximal chain in Li containing a + {3. Let He = { v E 'R : 8 E C => v ( 8) = 0} be the associated minimal
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60 prime subgroup. Then X/3 E He, yet Xa X/3 = Xa+/3 ff. He. Thus He is not an ideal and hence n is not an /ring by [BKW,9.1.2). Definition 3.3.5. An J 0 system satisfying the following is calle,d an /system: 1. If a and fJ are comparable, then (a, (J), ((J, a) EA. 2. If is maximal, then (8, ), (, 8), (, EA and 8 + = + 8 = 8 for every 8 In particular, + = We will shortly see that these additional assumptions on a bounded above / 0 system make maximal chains in t::,. into monoids. This is quite useful in our setting. For instance, it is not difficult to figure out when the /rings have a multiplicative identity, under the /system assumption. Proposition 3.3.6. Let (t::., +) be a bounded above /system. and 'R, each have a twosided multiplicative identity if and only if l::,. has a finite number of connected components Proof: Let (t::., +) contain only a finite number of connected components with maximal elements {, ... n}Let x.; be the characteristic function on the set {;} and let e = r:;;= 1 Xi Then for v E 'R, and 8 Et::., v e(8) = L v(a)e(fJ) = v(8)e(k) = v(8), a+/3=6 where k 8 is maximal. Likewise, e v(8) = L e(a)v(fJ) = e(k)v(8) = v(8). a+/3=6 Thus e is a twosided multiplicative identity. Conversely, assume that e E 'R, is an identity and let E t::,. be maximal with characteristic function Xw Then 1 = = e = e(). Thus, supp(e) has
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61 as many maximal elements as there are maximal elements in A. Since e E 'R, we must conclude that there are only finitely many connected components in A. It is handy to have the following definition: Definition 3.3. 7. Call an rsystem (A,+) unital if the ring 'R, has a multiplicative identity. The rings !Rand 'R, are semiprime, as we will now show. Thus the minimal prime subgroups are also prime ideals, by [BKW, 9.3.1]. Proposition 3.3.8. Let ( A, +) be an / 0 system. Then 'R, is semiprime. Proof: Let u E 'R, where u = L; eI a; Xo; for an index set I of supp ( u) and for a; E JR\ 0 for all j E J. Then U*u(o) = Lo=oi+o; aia;. Let oi be maximal in supp(u). We show that oi + oi is maximal in supp(u u). Let 8 = 8; + Ok E supp(u u). If 8; II oi, then by the / 0 system condition, 811 oi + OiOn the other hand, if 8;, Ok Oi, then 6 6i + 6i, as desired. Thus u u( Oi + oi) = a1 > 0 and we conclude that 'R, is semiprime. Corollary 3.3.9. Let (A,+) be an / 0 system. Then maximal chains in A are closed under the operation +. Proof: Let C ~Abe a maximal chain and let He= {v E 'R: 8 EC=} v(o) = O} be the associated minimal prime subgroup. Let a, fJ E C and assume by way of contradiction that a+ fJ iC. Then Xa XfJ = Xa+fJ E He but Xa, XfJ iHe. Thus He is not a prime ideal. This contradicts Proposition 3.3.8, by [BKW, 9.3.1]. One should ask if the maximal ideals of 'R are actually the maximal ideals; or equivalently, one should ask if 'R has the bounded inversion property. The answer is yes, if (A,+) is a unital /system.
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62 Proposition 3.3.10. Let(~,+) be a bounded above unital /system. An element u E 'R, is a multiplicative unit if and only if u (/. V for all maximal E ~Thus R satisfies the bounded inversion property. Proof: Let S be the set of maximal elements of and let e = LES X be the multiplicative identity. If u E 'R, is a multiplicative unit, then there exists v E 'R, such that u v = e. Hence 1 = e() = for every ES. Thus u() =/0 for all ES and hence u (/. V for all maximal E ~, as desired. Assume that u 8 E supp(u) and assume that v(,) is defined for all,> <5. Then u(a)v(T) is a summand of u v(8) only if <5 a, TE supp(u). Thus v(<5) is the only unknown in and is the unique solution of the equation O = u v ( <5). Hence, since if e is the multiplicative identity of 'R, and u e then u
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63 4> M K;z, = {g E G: all values of g are in 4>}. Proposition 3.3.12. Let (~, +) be a bounded above /system. Any closed convex subgroup of 'R is an ideal of 'R. Proof: Let K be a closed convex subgroup of 'R; let u E K+ and v E 'R. If u is a unit, then 4>K = r(G) and by the preceding theorem, K = G. So assume that u is a nonunit. Let E be maximal and assume that u v(,) = 0 for all~,> 8 and u*v(8) #0 for some 8. By the preceding theorem, we need to show that 8 E 4>K Let 8 =a+ (3 such that u(a)v(/3) #0. Then there is a maximal a' a such that u( a') #0. Then a' = a' + a' + fJ a + fJ = 8 and a' E 4> K since u E K. Therefore, we conclude that 8 E 4> K since 4> K is an order ideal. Hence, u v E K by the preceding theorem. We now seek the prime ideals and zideals among the prime subgroups V 0 and their associated value covers. The cover of V 0 is the set P 0 = { v E 'R : v(,) = 0 for all > 8} and is the smallest convex subgroup properly containing and Xo First let us recall the most general definition of zideal. Let G be a vector lattice, v E a+, and let {gn}~=l G be a sequence. Recall that the sequence converges relatively uniformly to g E G along the regulator v and write gn g, if for every c > 0 there exists Ne > 0 such that for all n Ne, we have that lg gnl cv. Let H be a convex subgroup (subvector lattice) of G. The pseudoclosure of H is H' = {g E G : 3{gn}~=l H, gn g for some v E n+}. Then His relatively uniformly closed if H = H'; let H denote the relative uniform closure of H. Then if G(g) denotes the convex subgroup (subvector lattice) of G generated by g, we define a convex subgroup (subvector lattice) H to be an abstract zideal if
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64 h E H,g E G and G(g) = G(h) imply that g EH. In fact, [HudPI, 3.4) says that this definition is equivalent to G(h) c H for all h E H. Proposition 3.3.13. Let(~,+) be an !system. If E is maximal, then V is a prime ideal of 'R which is an abstract zideal. Proof: Let u v E V, and u r/. V, Assume without loss of generality that u+, v+ E V and ur/. V, We show that vE V, Since ur/. V, we have that u() =I0. Thus since u+, v+ E V, 0 = u v() = L u(o:)v(,8) = L (u+ u)(o:)(v+ v)(,a), a+/J= a+/J= and since is maximal, we have that 0 = u()v() and therefore, v() = 0. This gives that vE V and v E Vw Hence V is a prime ideal. Since 'R/V is isomorphic to lR via the evaluation map u u(), we have that V is uniformly closed by [HudPI, 2.1). Thus by [HudPI, 3.4), V is an abstract zideal. Corollary 3.3.14. If(~,+) is a bounded above unital !system, then the maximal ideals are given by {V : E is maximal}. Proposition 3.3.15. Let(~,+) be an !system and let 6 E be nonmaximal. Define Po= { v E 'R: v('y) = 0 for all > 8}. Then P 0 is an abstract zideal and it is a prime ideal if and only if o: + ,8 > 8 for all o:, ,8 > 8. Proof: Let v E Pf Then there exists { vn}~ 1 Po such that Vn v, for some w E 'R,+. Let,> 8, then for every c > 0 we have that jvj(,) cw('y). Thus v(,) = 0 and Po is relatively uniformly closed. Therefore, P 0 is a zideal by [HudPI, 3.4). Assume that Po is a prime ideal and that there exist o:, ,8 > 6 such that o: + ,8 8. Then Xa+/J E Po and Xa+/J = Xa X/J But Xa, X/J r/. P 0 which is a contradiction.
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65 Conversely, assume that o+/3 > 8 for all nonmaximal o, /3 > 8. Let U*V E Po and assume, by way of contradiction, that u, v P 0 Then u v ( 'Y) = 0 for all 'Y > 8 and there exist elements o E supp(u) and /3 E supp(v), each maximal in the support set and such that o, {3 > 8. Assume, without loss of generality that o {3. If o' o, {3' {3 and at least one of the inequalities is strict, then o' + {3' < o + {3 and we conclude that u*v(o+/3) = u(o)v({3)+u({3)v(o). If o = {3, then since o+/3 > 8, we have that O = u v(o + {3) = 2u(o)v(f3) 10, a contradiction. If o > {3, then v(o) = 0 and hence O = u v(o + {3) = u(o)v(/3) 10, another contradiction. Thus we conclude that either u or v is in P 0 and therefore P 0 is a prime ideal. Let S be a totallyordered set. Recall that Sis an T/1set if whenever A, B S are countable and A < B, then there exists c E S such that A < c < B. Since JR is not an 'f/ 1 set, the ring 'R, is never an 'f/ 1 set. To see this, let 8 E /:1 and consider the sets {x 0 } > {(1 0 : n EN}. But, 'R, can satisfy a related, slightly weaker condition. Definition 3.3.16. We call a totallyordere,d set S an almost 'f/ 1 set if A, B S are countable and if A < B, then there exists c E S such that A c B. Note that R. is such a set. Proposition 3.3.17. Let ( !:l, +) be a totallyordered / 0 system. 'R, is an almost 'f/1 set if and only if !:l is an T/1 set. Proof: Assume that 'R, is an almost 'f/ 1 set. We first note that /:1 contains no suc cessor pair. Leto< {3 be a successor pair. Then the sets {nxa}nEN and {l/nx,0}nEN contradict the almost 'f/ 1 set condition. Let A = { oi}iEN, B = {/3; };EN C /:1, where 01 < 02 < < /3,i < f3i. Then Xa 1 < Xa 2 < < X,8 2 < X.8 1 and there exists / E 'R, such that Xa. f X.8; for all i,j. Let 'Y be the maximal element of supp(!). Then / Xai 0 implies that 'Y = Oi and /('Y) 1 or "f > Oi and /('Y) > 0. If 'Y = oi, then / Xa.+i < 0, which
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66 is a contradiction. Thus 'Y > a; for all i. Similarly, 'Y < /3; for all j. Therefore, is an 111set. Conversely, assume that is an 111set and let Ii < h < < 92 < 91 E 'R. Let 1 2 'Y 2 'Yi E be the corresponding maximal elements of the support sets. Let 4> = {;};eN and r = {'YiheN We have a few cases to consider: 4> and r are the same constant sequence, one of the sequences is eventually constant, or neither sequence is eventually constant. If there exists n E N such that ; = a = 'Yi for i, j > n, then we get the following string of inequalities in R: /i(1) /2(2) .. Y2h2) Y1h1), Since JR is an almost 111set, there is r E :R such that !;(;) r Yi('Yi) for all i,j. Then J; rx 0 < Yi for all i, j. If 4> is eventually constant and r is not, say ; = a for all i > n, for some n, then, by hypothesis, there exists /3 E such that a< /3 < r. Then /; < X/J Yi for all i, j. If neither sequence is eventually constant, then by the 111set hypothesis, there exists /3 such that ; < /3 < 'Yi for all i and /; < X/J < 9i for all i,j. 3.4 Survaluation Ring and n th Root Closed Conditions Recall that a commutative ring A is a survaluation ring ( or SVring) if A/ P is a valuation ring for every prime ideal P. In this section, we set down a character ization of those /systems which give SYrings. Let (~, <, +, be a totally ordered and cancellative abelian monoid with identity element We define the group of differences, q~, of as it is done in [Fu, X.4). Define an equivalence relation on~ x by (61, 6 2 ) ~ {'Yi, 'Y 2 ) if and only if 61 + 'Y2 = 'Y1 + 62. It is clear that the relation is reflexive and symmetric; the transitivity follows from the cancellation property of the monoid. We let q~ be the quotient x ~/ ~, denote the class of the element (6 1 6 2 ) by [6 1 6 2 ), and define an operation+ as is usual. That is, [61 6 2 ) + ['Y 1 'Y 2 ] = [(6 1 +'Yi)(6 2 + 'Y 2 )]. The
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67 cancellation in the monoid ensures that the operation is welldefined. The element [ ] is an identity and [6 2 6i] is an additive inverse of [61 62]. We define [61 62] ['Yi 12] if and only if 61 + '}'2 '}'1 + 62. By [Fu, X.4.4), this is the unique order on q~ which extends the order on ~Finally, oembed in q~ via 6 [6 ]. Let H be a partially ordered groupoid. Then h E H is called negative if hx x or xh x, or both for all x E H. The groupoid H is called negatively ordered if every element is negative. Definition 3.4.1. Let be a partially ordered semigroup. is called inversely naturally ordered if it is negatively ordered and 6 < a implies that there exists /3 E such that 6 = a + {3. Example 3.4.2. Let~= {1 }~= 1 U [1,oo) CR. be inversely ordered with the usual addition in the reals. Then is an !system which is not inversely naturally ordered. To see this, note that (1 + c = 1 if and only if c = n = 2 or c=l,n=l. Let (~, +,)be a totallyordered abelian cancellative monoid with maximal element, such that(~,+) is an !system. Let X = {x 6 : 6 E ~}Then (X,*,X) is a totallyordered abelian monoid which is isomorphic to (~, +, ). Since X is written multiplicatively, the elements of the group qX are quotients and we denote them as such in the following proof. Theorem 3.4.3. Let(~,+,) be a totallyordered abelian cancellative monoid with maximal element Let X = {x 0 : 6 E ~} and 'R, = F(~, R.). The following are equivalent: 1. 'R, is a valuation ring. 2. 'R, is Iconvex.
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68 S. 'R, is Bezout. 4'R, is convex in q'R,. 5. X is convex in qX. 6. D. is convex in qD... 7. D. is inversely naturally ordered. Proof: The equivalence of (1), (2), and (3) is [MW, Theorem 1]; the equivalence of (2) and (4) is [ChDi2, Lemma 2]. (2) :::} (5) : Let x 7 x 0 /x13 Xo Then O Xo XH/3 and by (2), there exists / E 'R, such that Xo = f Xo+P If f = E;EJ /;X; else Thus, for some j E J, we have that a= (3 + 6 +;,I;= 1 and fk = 0 for all k/= j. Hence f = X
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69 Let w = LkEK bkx 7 ,.. Then w() = b 1 > 0 and hence b\ w is invertible in n. Let x = L;o a;x 61 Then by cancellation in .6., we have that 0 x w and 0 b\ x b\ w. If we let f = (b\ w)1 (b\ x), then b\ x = b\ w f and x = w f. Finally, u = X{Ji x = X{Ji w f = v f, as desired. The following lemma is wellknown and routine to verify. Lemma 3.4.4. Let (.6., +) be an system and C .6., a maximal chain. Denote the associated minimal prime subgroup by He= {u En: C n supp(u) = 0}. The map cp : F(.6., lR) + F(C, lR) given by restriction is a surjective lring homomorphism with kernel He. Thus, Rf He~ F(C,R) Corollary 3.4.5. Let ( .6., +) be a 1.mital !system. Then n is an SVring if and only if each maximal chain in .6. is inversely naturally ordered. Example 3.4.6. 1. If .6.1 = (0, oo) c lR is inversely ordered with the usual addition of real numbers, then F(.6. 1 lR) is an SYring. 2. If .6.2 = {1 }~=l U [1, oo) c JR is inversely ordered with the usual addition in the reals, then F(.6. 2 JR) is not an SYring. 3. Let A= JR[[x, y]] be the ring of formal power series in the indeterminates x, y. Order the monomials lexicographically via 1 x, y and xiyi < xkyl if and only if k < i or k = i and l < j. The ring A is not an SYring since it is not 1convex: note that O y x and the equation y = xf has no solution f EA. Let Zx = Zy = { n E Z : n O} be inversely ordered. In the lexorder described above, if .6. = Zx x Zy, then A~ F(.6., JR). We convexify A in qA by convexifying .6. in q.6.. That is, if .6.c = .6. U {(n, m) E Z x Z : n > 0, m < 0} then F(.6.c, JR) ~ A( { xiyi : i > 0 or i = 0 and j > 0}) U A is an SYring which is the convexification of A in qA.
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70 Recall that an fring A is n th root closed if for every a E A+ there exists b E A such that a = bn. This property arises in 7l if there is a certain amount of divisibility in the arithmetic structure of ~Definition 3.4. 7. Let ( ~, +) be an rsystem. is called ndivisible if for every 8 E there exists a E such that na = 8. We say that the system is divisible if it is ndivisible for all n E N. Proposition 3.4.8. Let ( ~, +) be a totally ordered !system such that 1l is n th root closed. Then is ndivisible. Proof: Let 8 E ~Then Xo = vn for some v = E;o a;Xa;If 8 = is maximal in ~, then 8 = n, so we assume that 8 :/. If a 1 E supp(v) is maximal in the support set, then na 1 E supp(vn) is maximal. Therefore 8 = nai, as desired. Proposition 3.4.9. Let(~,+) be a totally ordered inversely naturally ordered! system with maximal element. If~ is ndivisible, then 1l is n th root closed. Proof: We begin with squareroots Let u En,+ be given by u = E;EJ a;Xa ; for some index set J, a; E JR, and a; E for all j E J. If u() :/0, then we define a squareroot v recursively on ~To begin, let v() = v1u(ji') and assume that a 1 is maximal in supp(u) \ {}. Let v(8) = 0 for all a1 < 8 < Then we define v(ai) = u(ai)/2v(). Let S be the Nlinear span of supp(u) and define v(8) = 0 for all 8 (j. S. If v(,) is defined for all > 8, then we define v(8) to be the unique solution of the equation u(8) = Eo+r= 6 v(a)v(r), where, necessarily, a, r E { a E S : a 8}. Then u = v v, by construction. Now, assume that u() = 0 and let a 1 be the maximal element in the support of u. Since~ is inversely naturally ordered, for every j E J, there exists 8; E such that a;= a1 +8;. Note that 81 =.Then u = Xai E;EJ a;Xo; Let w = E;EJ a;Xo;, then w() :/0. Thus w = v 1 v 1 by the above construction. Since is 2divisible,
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71 a 1 = 2 11 for some 11 ED.. Therefore x 01 = X,y 1 X,y 1 Letting v = X,y 1 vi, we then have that v v = X,y 1 v 1 X,y 1 v 1 = x 01 w = u. Thus n th roots exist when n is a power of 2. Let n be odd and let v E 'R, be given by v = EkEK bkXp,. for some index set K, bk E R, and f3k E D. for all k E K. As with square roots we consider two cases. First assume that v() =/0 and define an n th root recursively. Let w() be a real n th root of v() and let v(c5) = 0 for all /3 1 < 8 < where /3i is the maximal element of supp( v) \ {}. Then the nfold convolution product equation we must solve reduces to v(/3 1 ) = wn(/3 1 ) = n(w())nlw(/3 1 ). In order to see this, we proceed by induction on n. If n = 2, then W*w(/31) = w()w(f31)+w(/31)w() = 2w()w(/31) Assume that the statement holds for n = m. Then wm+ 1 (/31) = (w wm)(/31) = w()wm(/3i) + w(/31)wm() = w()m(w())m1 w(/3 1 ) + w(f31)(w())m = (m + l)(w())mw(/3i) as desired. We may then define w(/3 1 ) = v(/3 1 )/(n(w())n1 ). In general, we let w(,) = 0 if 'Y is not in the Nlinear span of the support of v Assume that w(,) is defined for all,> 8. We show that the equation v(c5) = wn(a) is linear in w(8); hence, we may define w(8) to be the unique solution of this equation. H n = 2, then w 2 (8) = Eu+T=o w(o)w(r) + 2w(8)w(). Assume that wm(a) is linear O',Tf.0 in w(c5). Then wm+l(a) = L w(a)wm(r) + w()wm(8) + w(8)wm() O'+T=6 O',Tf.0 is linear in w(8), by induction, since w(8) will not appear in wm(r), as 8 < 'T and D. is an !system. Thus, in this case, v has an n th root. Second, assume that v() = 0 and proceed as in the squareroot case. Let /3 1 be the maximal element in the support of v. Since D. is inversely naturally ordered, for every k E K, there exists 8k E D. such that f3k = /3 1 + 8k. Note that 8 1 = Then
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72 v = XtJi LkEK bkX5,.. Let x = LkEK bkX5,., then x() /:0. Thus x = wf, for some w 1 by the above construction. Since 6. is ndivisible, /31 = ny 1 for some 'Y i E 6.. Therefore Xf3 1 = (x, 1 )'1'. Let w = x, 1 w1; then w" = (x, 1 w1)" = Xf3 1 x = v. Corollary 3.4.10. Let (6., +) be a totally ordered inversely naturally ordered! system with maximal element 6. is divisible if and only if 1l is n th root closed for all n. Example 3.4.11. 1. Let 6. 1 = [0, oo) c JR be inversely ordered with the usual addition of the real numbers. Then F(6.1, R) is n th root clo.5ed for all n. 2. Let 6.2 = {n E Z : n 2:: 0} be inversely ordered. Then 6.2 is not 2divisible and Xi > 0 has no squareroot. That is, if Xi = v 2 then v(O) = 0 and we then conclude that 1 = x1(l) = 2v(0)v(l) = 0, a contradiction. 3. Let 6. 3 = {l}~=3U [ 1, oo) CR be inversely ordered with the usual addition in the reals, then 6. 3 is not inversely naturally ordered and similarly x 1 has no squareroot since 1 has no nonzero summand. Proposition 3.4.12. Let (6., +) be a totally ordered /system with maximal ele ment Then if has an immediate predecessor 1r, then X1r has no squareroot. If there exists > 6 E 6. such that 6 has no nonmaximal summand other than itself, then X5 has no squareroot. Based on the preceding examples and results on n th roots, we formulate the following: Conjecture 3.4.13. Let (6. +) be a totally ordered /system with maximal element F(A, R) is squareroot closed if and only if 6. is 2divisible and every nonmaximal element of 6. has a nonmaximal summand other than itself
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73 Recall that a field K is realclosed if every positive element is a square and every polynomial p E K[x] of odd degree has a root in K. An integral domain R is called realclosed if qR is a realclosed field. Let (~, +) be a totally ordered inversely naturally ordered !system with maximal element Assume that is also 2divisible. Then 'R, is a 1convex and squareroot closed !domain. By [ChDil, Theorem 1], under these conditions, 'R, is realclosed if and only if every odd degree polynomial over 'R, has a root. What additional assumptions on are necessary to guarantee the realclosed property? Conjecture 3.4.14. Let (~, +) be a totally ordered commutative !system with maximal element If~ is divisible and inversely naturally ordered, then 'R, is real closed We end this section by shedding a little light on this conjecture. Recall from [HLM] that a commutative fring A with 1 satisfies the Intermediate Value Theorem for polynomials (or is an !VTring, for short), if for every p(t) E A[t], and pair of distinct elements u, v E A such that p( u) > 0 and p( v) < 0, there exists w E A such that p(w) = 0 and u I\ v w u V v. We show that a totally ordered commutative semiprime valuation !domain with identity is realclosed if and only if it is an IVT ring. It is necessary to record the following unpublished theorem of Suzanne Larson, which was communicated via electronic mail on April 17, 1997. Her proof follows. Theorem 3.4.15. Let A be a commutative semiprime !VTring with identity. If S is a multiplicatively closed subset of regular elements of A+, then the ring of quotients, s1 A is an !VTring. Proof: Let p(t) E s1 A[t] be given by p(t) = a 0 b 0 1 + a 1 b 1 1 t + + anb;/tn and assume that p( u1 v 1 1 ) 0 and p( u 2 v 2 1 ) 0. Let d = vfv 2 b 0 b 1 bn. Then d E S is regular Define a new polynomial q(t) E A[t] by
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74 Then q(u 1 v 2 ) = dp(u 1 v 1 1 ) 0 and q(u 2 v 1 ) = dp(u2v 2 1 ) ::; 0. Since A is an IVTring, there exists w E A such that u 1 v 2 I\ u 2 v 1 ::; w ::; u1 v2 V U2V1 and q( w) = 0. Then u 1 v 1 1 /\ u 2 v 2 1 ::; wv 1 1 v 2 1 ::; u 1 v 1 1 Vu2v 2 1, and dp(wv 1 1 v 2 1 ) = q(w) = 0. Since dis regular, p(wv 1 1 v 2 1 ) = 0 and we conclude that the quotient ring is an IVTring. Proposition 3.4.16. Let A be a totally ordered commutative semiprime valuation !domain with identity. Then A is realclosed if and only if it is an /VTring Proof: If A is realclosed, then qA is a realclosed field and, by [ChDi2], qA is an !VTfield. Let p( t) E A[t] be such that p( u) > 0 and p( v) < 0, for some u, v E A. Then there exists w E qA such that p(w) = 0 and u I\ v ::; w ::; u V v. Since A is a valuation domain, A is convex in qA by [ChDi2, Lemma 2]. Hence, w E A and we have that A is an IVTring. Conversely, if A is an IVTring, then qA is an !VTfield, by the preceding theorem of Larson. Then, by [ChDi2], qA is realclosed and therefore A is realclosed.
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CHAPTER 4 RAMIFIED PRIME IDEALS In this chapter we expand on the notion of a ramified prime ideal, as defined in [Le], which we introduced in Chapter 2. We first examine the concept in general and then move to try to understand ramified maximal ideals which correspond to nonisolated G 6 points. This result is then used to consider local versus global ramification conditions. 4.1 Ramified Points Definition 4.1.1. Let X be a completely regular space. A prime ideal of C(X) is ramified if it is the sum of the minimal prime ideals that it contains. We define p E X to be ramified if Mp is ramified. A point p E X is totally ramified if every prime zideal contained in Mp is ramified. The space X is (totally) ramified if every nonisolated point of X is {totally} ramified. A ramified ideal of C(X) is a prime ideal, by Corollary 2.2.6. LeDonne proves that a ramified prime ideal of C(X) is necessarily a zideal. Let us consider two extreme conditions. Recall that we say a point p E X is an Fpoint if Op is prime. If pis an Fpoint, then since Op is the unique minimal prime ideal contained in Mp, Mp is not ramified Likewise, in this case, no prime zideal contained in MP is ramified. On the other hand, the condition of total ramification ensures branching at every prime zideal. Analytically, LeDonne shows [Le, 3]: Theorem 4.1.2. If Xis a metric space then every maximal ideal of C(X) is totally ramified. 75
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76 Note that this result says that every maximal ideal of C(X) (fixed or free) is ramified, if X is metric. We do not know of any weaker topological condition which guarantees total ramification of C ( X). Definition 4.1.3. Let A be a commutative /ring with identity. For any integer n 2, call a prime lideal P nlimbed if P is the sum of n noncomparable prime lideals of A which are properly contained in P. A point p of X is nlimbed if M 11 is nlimbed. Note that any nlimbed lideal P is necessarily ramified and rk{P) n. Example 4.1.4. We now present an example of an /ring 'R, in which a maximal ideal is ramified but not nlimbed for any n. Let Ao = [O, oo) R and define A~ = (1/n, oo) JR, for i = 1, 2, where each interval is inversely ordered. Let An = A~ II~ be the disjoint union and then let A = 6,o II (IInEwAn). We obtain a root system with the induced ordering which we describe in the following diagram: .L\o {4.1) We endow A with a partial commutative associative binary operation. Let ( )~ denote the sum in parentheses as the usual sum of real numbers residing in Ai n,
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77 the mark "" signifies that the sum is undefined. Note, to conserve space, the table is completed by reflection across the diagonal. + r E Ao r EA:_ r EA:,. r E At r E Af s E Ao (r + s)o (r+s):, (r + s)!.. (r + s)t (r + s)f II E A:, (r + s); (r + s); (r + s)t ifk < n; else (r + s)t if l < n; else s E A:,. (r + s):,. (r + s)Z if k < m; else (r + s)j if l < m; else s EA! (r + s)t s E Aj (r + s)t Let 'R, = F(fl., R) and Vn = Llo II (IIm>nA~); let Co = Ao II (llnEwA~) and Cn = Vn II A!. Then the minimal prime ideals of 'R, correspond to these maximal chains and are given by Qn = { u E 'R, : u(Cn) = {O}}, where n = 0, 1, 2, .... Any supremum vjEJ Q; over a finite set J C w is the prime ideal Pm given by { u E 'R, : u(Vm) = {O}}, where m is the maximum element of J. Hence, for all n E w, the maximal ideal Yo is not nlimbed since it is not a finite supremum of minimal prime ideals. However, V 0 = V nEw Qn, and so it is ramified. We show in Proposition 2.3.4 that for any nonisolated point p of a first countable space, the maximal ideal MP is nlimbed for every n. H the space is also cozerocomplemented, then rk(Mp) 2:: 2' and Mp is 2'limbed. From this, we also obtain the following, which is weaker than Theorem 4.1.2: Corollary 4.1.5. Every metric space is ramified. The following theorem means that if X is a metric space that is not pseudocompact, then there exist points of f3X \ X such that MP is minimal. Hence, not every maximal ideal of C(X) branches nontrivially in the root system Spec(C(X)). Recall that we call a topological space X perfect if every closed set of Xis a G 6 set. Note that any metric space is perfect. A point of /3X is remote if it is not in the /JXclosure of any nowhere dense subset of X. A point p E /3X \ X is a Cpoint if p intpx\x(clpxZ(/) n /3X \ X) for all / E C(X). A theorem similar to the following appears in [W, 4.40]. All the proofs there carry through here, verbatim, under our reduced hypotheses.
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78 Theorem 4.1.6. Let X be a completely regular space and consider the following conditions on a point p E {3X \ X: 1. pis a Cpoint. 2. Z[MP] contains no nowhere dense set. s. MP= QP. ,f.. p is a remote point Without additional assumptions, (3) => (1). Let X be perfect and assume the exis tence of a remote point, then (4) => (2) => (3). If X is perfect and the set of isolated points of X has compact closure in X, then (2) => (4). If X is realcompact and Cpoints exist, then (1) => (2). It is not known if a remote point p always has the property that MP = QP. We do know the following about the rank of a remote point: Proposition 4.1.7. Let X be a completely regular space. Let p E {3X be a remote point. Then rkc(x) (MP) = 1. Proof: In [vD, 5.2], it is demonstrated that no remote point is in the closure of two disjoint open sets of {3X. Thus, in particular, no remote point is in the closure of two disjoint cozerosets of {3X. By [HLMW, 3.1], we have 1 = rktJx(p) = rkc(tJX)(M*P). Finally, since C(f3X) is rigid in C(X), we have rkc(x)(MP) = 1. Finally, we ask: does ramification of a point in X indicate ramification in {3X, or vice versa? Proposition 4.1.8. A point p EX is ramified in X if and only if it is ramified in {3X. Likewise, a point p E vX is ramified in vX if and only if it is ramified in {3X. Proof: This is a corollary of Proposition 2.2.12, since we know C(f3X) = C*(X) and MfX = M{ n C(f3X).
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79 4.2 Ramilied G 6 points The main theorem, Theorem 4.2.5, of this section provides a good method for checking the ramification of G 6 points. We will use it to characterize ramified G 6 points in normal countably tight spaces and to find some ramified points in product spaces. We first discuss the following proposition, which we will obtain as a corollary to Theorem 4.2.5. Proposition 4.2.1. Let p E X be a G 6 point. If X \ p is not c embedded in X, then rk(p) 2. Since an Fpoint has rank 1, the preceding proposition, proved in [Le) and (in greater generality) by van Douwen in [vD), shows that a G 0 point, p, is not an Fpoint if it has the property that X \pis not C*embedded in X. We give a counterexample for the converse if the G 0 condition is lifted. Example 4.2.2. Let X = B 2 (N, (3N \ N) be as defined in Example 5.2.5. There, we show that there exists a point p of the corona such that rk(p) = 2, X \ p is C*embedded in X and pis not ramified. But no point of the corona is a G 6 The following two results are Theorems 2.1 and 2.2 of [K2); we will use these to prove our theorem on the ramification of G 0 points. Theorem 4.2.3. Let p be a nonisolated G 0 point of X. If Z E Z[C(X \ p)] then clx(Z) E Z[C(X)). Define : Z[C(X \ p)] + Z[C(X)] by ,(Z) = clx(Z). Let be the extension of the identity map X \p+ X to the largest subspace X 1 f3(X \p) such that it is extendible as a continuous map into X.
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80 Theorem 4.2.4. Let p be a nonisolated G 0 point of X then 1. The mapping 1 is onetoone from the set of prime zfilters on X \p converging to points of 4>1 ({p}) onto the set of prime zfilters on X contained properly in Z[Mp]2. A prime zfilter W on X \p converging to a point of 4>1 ( {p}) is a zultrafilter if and only if ,(W) is maximal in the class of prime zfilters on X contained properly in Z[Mp]Theorem 4.2.5. Let p be a nonisolated G 0 point of X. The point p is ramified if and only if X \ p is not C* embedded in X. Proof: Let p be a nonisolated G 0 point of X. If p is not ramified then the prime zideal P = EQEMin(Mp} Q is properly contained in Mp. We will show, in this case, that every point of Xis the limit of a unique zultrafilter on X \p. Then by [GJ, 6.4], X \pis C*embedded in X. Let q E X \ p. Then Mq E Max(C(X \ p)) gives rise to the zultrafilter Uq = Z[Mq] on X \p. Clearly q E n{clx(U): U E Uq}The uniqueness of Uq is a standard result [GJ, 3.18]. By (GJ, 6.3(b)], there exists a zultrafilterU on X\p converging top. Assume that there exists another such zultrafilter, V. Let U = ,U and V = ,V. Then Qu = ZtU and Qv = ZtV are prime zideals of C(X) which are properly contained in Mp. If Qu P then U = Zx[Qu] Zx[P]. Hence U = ,1 u ,1 Zx[P], which contradicts that U is a zultrafilter on X \p. Likewise, Qv is not properly contained in P. Thus P Qu, P Qv and by [GJ, 14.8(a)], we must have either Qu c Qv ~ ~ or Qv C Qu. But Qu c Qv gives that U V and therefore U V. Since U is an ultra.filter, U = V, as desired. In a similar manner, if Qv C Qu, then V = U. Conversely, assume that p is ramified. Then Mp = E Min(Mp) and there exists more than one prime zideal in C(X) which is maximal in the class of prime
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81 zideals properly contained in Mp. These give distinct prime zfilters on X which are maximal in the class of prime zfilters on X properly contained in Z[Mp]Hence, via,, we have distinct ultrafilters on X \ p converging top. Again by (GJ, 6.4), X \ p is not C* embedded in X. Corollary 4.2.6. If X is a metric space, then X is ramified. Corollary 4.2. 7. If X is first countable and p E X is nonisolated, then X \ p is not C* embedded in X. Corollary 4.2.8. If p EX is a G 0 point and X \pis not C*embedded in X, then rk(p) 2. Let X and Y be completely regular spaces which are not Pspaces and let W =Xx Y. We conjecture that every nonisolated point of Wis ramified. We use Theorem 4.2.5 to deduce two partial answers to this question. Proposition 4.2.9. Let X and Y be completely regular spaces and let W =Xx Y. Let x E X and y E Y be nonisolated G 0 points and let p = (x, y) E W. Then W \ {p} is not C* embedded in W. Proof: Since X is completely regular and x is a G 0 point of X, { x} is a zeroset of X. Say, {x} = Zx(f) for some f E C(X). Then we have for Ex= {x} x Y, Ex= {x} x Y = 1rt(Zx(/)) = Zw(/ o 1rx) where 1rx denotes the natural projection from W onto X. Likewise, we have that E 11 =Xx {y} = Zw(g o 1ry) where g E C(Y) such that Z 11 (g) = {y} and 1ry is the natural projection from W onto Y. Let f and g denote the restrictions off 01rx and go1ry to W\ {p}. Then we have that Ex\ {p} = Zw\{p}(f) and E 11 \ {p} = Zw\{p}(g) are disjoint zerosets of W\ {p}. Thus Ex\ {p} and E 11 \ {p} are completely separated in W\ {p}. But p E clw(Ex \ {p} )nclw(E 11 \ {p}) and therefore Ex\ {p} and E 11 \ {p}
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82 are not completely separated in W. By the Urysohn Extension Theorem, W \ {p} is not c embedded in W. Corollary 4.2.10. Let X and Y be completely regular spaces and let W =Xx Y. Let x EX and y E Y be nonisolated G 0 points and let p = (x, y) E W. Then p is ramified in W Proof: Follows from Theorem 4.2.5. Proposition 4.2.11. Let X and Y be completely regular spaces and let W = Xx Y. Let p = ( x, y) E W be non isolated. If W \ {p} is normal, then W \ {p} is not c embedded in W. Proof: Let Ex = { x} x Y and E 11 = X x {y}. Then Ex\ {p} and E 11 \ {p} are disjoint closed sets in the normal space W \ {p}. Thus, they are completely separated in W\ {p}. But p E clw(Ex \ {p}) n clw(E 11 \ {p}), so Ex\ {p} and E 11 \ {p} are not completely separated in W. Therefore, by the Urysohn Extension Theorem, W\ {p} is not c embedded in W. Corollary 4.2.12. Let X and Y be completely regular spaces and let W =Xx Y. Let p = (x, y) E W be a nonisolated G 0 point of W. If W \ {p} is normal, then p is ramified in W Proof: Follows from Theorem 4.2.5. We now investigate the ramification of G 0 points in a class of spaces more general than metric or first countable spaces. A topological space X is countably tight if for a subset U c X we have that any p E cl(U) is in the closure of a countable set S c U. A FrechetUrysohn space is one in which every p E cl(U) is the limit of a sequence of distinct points {p,J c U. It is evident that any FrechetUrysohn space is countably tight.
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83 Lemma 4.2.13. Let X be a normal topological space and let p EX be nonisolated. Then X \ {p} is c embedded in X if and only if p cl x (A) n cl x ( B), whenever A and B are completely separated in X \ {p}. Proof: By the Urysohn Extension Theorem, X \ {p} is C*embedded in X if and only if A and B are completely separated in X, whenever A and B are completely separated in X \ {p}. Assume that X \ {p} is c embedded in X and let A and B be completely separated in X \ {p}. Then A and B are completely separated in X and hence are contained in disjoint closed sets of X. Thus p clx(A) n clx(B). Conversely, let A and B be completely separated in X \ {p}. We wish to show that A and Bare completely separated in X. By hypothesis, p
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84 Corollary 4.2.15. Let X be normal FrechetUrysohn space and let p E X be non isolated. Then X \ {p} is C* embedded in X if and only if there do not exist two sequences in X which are completely separated in X \ {p} and converge to p. Finally, Proposition 4.2.14 and Theorem 4.2.5 imply the following. Corollary 4.2.16. If X is a normal countably tight topological space and p EX is a nonisolated G 0 point, then pis ramified in X if and only if for every two countable sets S 1 and S 2 which are completely separated in X \ {p}, p clx(S1) n clx(S2)4.3 Ramification via CEmbedded Subspaces Let A and B be commutative rings with identity. Assume that we have a surjective ring homomorphism f: A B, with K = Ker(!). Recall that there is a onetoone orderpreserving correspondence between Spec(B) and the set of prime ideals PE Spec(A) such that K P. Let ME Max(B) be such that M = P1 + P2 for some nonmaximal proper primes P 1 P 2 E Spec(B). If NE Spec(A) corresponds to M, then we have B/M ~ (A/K)/(N/K) ~ A/N. So NE Max(A). Let Q 1 ,Q 2 be the prime ideals of A corresponding to Pi and P2. Then ( Qi + Q2) / K ~ P1 + P2 via the surjective map given by a + b f (a) + f (b) with kernel K. Thus, by the correspondence, we have that N/ K M = Pi +P2 ~ (Qi +Q2)/ Kand N = Qi +Q2. In fact, we have shown: Proposition 4.3.1. Let A and B be commutative rings with identity such that there exists a surjective ring homomorphism f : A B with K = Ker(!). If P B is a prime ideal which is a sum of two distinct prime ideals then there exists Q E Spec(A) such that Q/ K ~ P and Q is a sum of two distinct prime ideals of A. Ramification in a Cembedded subspace implies global ramification. Corollary 4.3.2. Let X be a completely regular space and let Y be a Cembedded subspace. If a point p of Y is 2limbed in Y, then p is 2limbed in X.
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85 Proof: Since Y is Cembedded in X, we have a surjective ring homomorphism from C(X) onto C(Y), given by restriction with kernel {f E C(X) : Y Zx(f)}. Hence this result follows from Proposition 4.3.1. Proposition 4.3.1 also gives the following, since any compact subspace of a completely regular space is Cembedded. Corollary 4.3.3. 1. Let X be compact, Y X a closed subspace If a point of Y is 2limbed in Y then it is 2limbed in X. 2. Let X be a compact space consisting of more than one point. If every noniso lated point in a proper zeroset of X is 2limbed, then every nonisolated point of X is 2limbed. Example 4.3.4. The Cantor Set is metric, hence every point is 2limbed. By [E, 3.12.12c], every point of 2r, the Cantor space of weight T, is contained in a closed set which is homeomorphic to the Cantor Set. Thus every nonisolated point of 2r is 2limbed. In fact, induction on Proposition 4.3.1 gives that every point of the Cantor space is nlimbed for every n EN. Conversely, if a maximal ideal of A containing K is a sum of primes containing K, then by the correspondence given above, its image is a maximal ideal which is a sum of primes in B. That is, if NE Max(A), K N, and N =Qi+ Q2 such that K Qi and K Q2, then f(N) E Max(B) as B/f(N) Or! (A/K)/(N/K) A/N. And f(N) = f(Qi + Q2) = f(Qi) + f(Q2) ~ Qi/K + Q2/K. This gives a partial converse: Proposition 4.3.5. Let A and B be commutative rings with identity such that there exists a surjective ring homomorphism f : A B with K = Ker(!). Let P be a prime ideal of A containing K. Then P is a sum of two distinct prime ideals
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86 containing K if and only if f(P)/ K is a prime ideal which is a sum of two distinct prime ideals of B. Corollary 4.3.6. The ringhomomorphic image of a commutative ramified ring with identity is ramified. Corollary 4.3.7. Let Y be a Cembedded subspace of X and let p E Y have finite rank in X. If p is ramified in Y, then p is ramified in X. If any function in C(X) vanishing on Y also vanishes on a neighborhood of p, then p is ramified in Y if and only if p is ramified in X. Proof: The first statement is an application of Proposition 4.3.1 by induction. The second statement follows from Proposition 4.3.5 by induction. Note that the hypothesis of the second statement of the above merely de mands that the kernel of the restriction map be contained in o;. This is satisfied if Y is open or if Y is a Pset. The preceding results indicate that ramification is a local property. Let A be a commutative ring with identity, let S be a multiplicative system in A such that 1 E S. Then there exists a onetoone correspondence from Spec(s1 A) onto {PE Spec(A) : P n S = 0}. The proofs of the following are routine: Proposition 4.3.8. Let A be a commutative ring with identity, let S be a multi plicative system in A such that 1 E S. 1. If P E Spec(s1 A) is a sum of two proper primes, then the preimage of P, the set {x EA: x/1 EA}, is a prime ideal which is a sum of two proper primes in A. 2. If Q E Spec(A) is a sum of primes and Q n S = 0, then s1 Q is a sum of primes in s1 A.
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87 Corollary 4.3.9. Let Y be a subspace of X such that C(X) <t C(Y) is a ring of fractions map. That is, there is a multiplicative system S C C(X) such that 1 E S and C(Y) = s1 (C(X)). Then: 1. For p E Y, rky(p) rkx(p). 2. Let p E Y. If M{ n S = 0, then there is PE Spec(C(X)) such that P M{ and rky(p) = rkx(P).
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CHAPTER 5 mQUASINORMAL fRJNGS In [Lal 3], Suzanne Larson defines the notion of a quasinormal !ring; one in which the sum of any two distinct minimal prime ideals is a maximal ideal or the entire fring. We generalize this definition and a few of her results. 5.1 Definitions Definition 5.1.1. Let A be a commutative fnng with identity and let m be a positive integer. Call A mquasinormal if the sum of any m distinct minimal prime ideals is a maximal ideal or the entire fri,ng A. If X is a topological space such that C(X) is m quasinormal, then we call X an Fmspace. Note that the "2quasinormal" is Larson's "quasinormal" condition, the "1quasinormal" condition is equivalent to von Neumann regularity, and if A is quasinormal then A is nquasinormal for any n > m. Hence, the F 1 spaces are exactly the Pspaces and any Fmspace is an Fnspace, when n > m. Theorem 5.1.5 generalizes [Lal, 3.3] and characterizes the mquasinormal semiprime !rings. Note that [Lal, 2.2], which we now state, gives necessary and sufficient conditions for a semiprime fring to have the property that the sum of any two distinct minimal prime ideals is a prime ideal. This condition is stronger than the assumption we make in our theorem, but this result indicates when one can expect to be able to apply it. Theorem 5.1.2 Let A be a semipri,me fring. The following are equivalent: 1 The sum of any two semiprime ideals is semiprime. 88
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89 2. The sum of any two minimal prime ideals is prime. S. The sum of any two prime ideals is prime. 4,. For any a, b E A+, the ideal a.1..1. + b.1..1. is semiprime. 5. For any two disjoint elements a, b E A+, the ideal a.l. + is semiprime. 6. For any a, b EA+, the ideal a.1. + b.1. is semiprime. 7. When x,a,b,c,d E A+,a,b =/ 0 are such that x 2 = c+d and a/\c = b/\d = 0, there exist g, h E A such that x = g + h and g I\ a = h I\ b = 0. The theorem above holds for C ( X) since the sum of two prime ideals is prime by [GJ, 14B]. We will use the following lemmas. The first follows from the fact that a prime ideal P of a semiprime /ring is minimal if and only if for every p E P there exists q P such that pq = 0. The second lemma shows the existence of certain functions which we will take for granted in the proofs to follow. Lemma 5.1.3. Let A be a semiprime /ring in which the sum of any m distinct minimal prime ideals is a prime ideal. Let P be a prime ideal. Then P is minimal with respect to containing EJ=i af if and only if for every p E P there exists q ff. P such that pq E Ei= 1 af. Lemma 5.1.4. Let A be a commutative /ring. Let n 2:: 2 and let Q 1 .. Qn be distinct minimal prime ideals. Then there exists an element f E Qt\ LJ#i Q;, for each i = 1, ... n. Proof: Let fk E Qt \ Qk for k = 1, ... n. Then, by convexity, we have that f = v:=2/i E Qt\ LJ#i Q;, as desired. Theorem 5.1.5. Let m be a positive integer. Let A be a commutative semiprime /ring with identity in which the sum of any m distinct minimal prime ideals is a prime ideal. The following are equivalent:
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90 1. A is mquasinormal. 2. For every nonmaximal prime ideal P, rk(P) m 1. 9. Let { a; }~ 1 be a family of positive pairwise disjoint elements of A. Proper prime ideals containing L.1=i af are maximal ideals. 4. Let { a;}~ 1 be a family of positive pairwise disjoint elements of A, let M be a maximal ideal containing the ideal LJ=l af, and let p E M. Then there exists z rl. M such that zp E L.1= 1 af. Proof: (1) => (2): Let P be a nonmaximal prime ideal of A such that rk(P) m and let Q 1 ... Qm be m distinct minimal prime ideals that are contained in P. Then LJ=l Q; P is not maximal, hence A is not mquasinormal. (2) => (3) : Let P be a prime ideal containing L.1=i af. Then af P for every j. Therefore, a; rl. O(P) for all j, and hence P contains at least m minimal prime ideals by the pigeonhole principle. Thus condition (2) gives us that Pis a maximal ideal. (3) # (4): Follows from Lemma 5.1.3. (3) => (1): Let M be a maximal ideal and let Q 1 ... Qm M be minimal prime ideals. Then by hypothesis, L.1=i Q; is a prime ideal and we are left to show that it is a maximal ideal. For each j = 1, ... m, let a; E Qt \ Uit;Qi and define b; = /\i#;ai/\~ 1 ak E L.1=i Q;. Then {b;}.i=l is a pairwise disjoint set of m distinct elements of L.1=i Q; and by the choice of the a;'s, we have that /\;:1= 1 ak E n;:1= 1 Qk by convexity and /\i#iai rl. Q; for each j. Hence b; rl. Q; and bf Q;, for each j. Thus L.1=i Q; is a maximal ideal by condition (3), since L.1=i bf~ L.1=i Q;. The quasinormal condition is a variation of the normal condition, which is that the sum of any two minimal prime ideals of a semiprime /ring with identity is the entire !ring. This is discussed in [Hu]. The expected generalized definition
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91 follows, along with a theorem recording two equivalent conditions. The result is a special case of Theorem 5.1.5 and the proof follows immediately from the one above and from [Hu, Theorem 8]. Definition 5.1.6. Let m 2 be a positive integer. An /ring A is mnormal if for any painnise disjoint family {a;}j=:: 1 we have that A= Ei=l af. Theorem 5.1. 7. Let A be a commutative semiprime !ring with identity and let m 2 be a positive integer. The following are equivalent: 1. A is mnormal. 2. For any maximal ideal M, we have that rk(M) m 1. 3. The sum of any m distinct minimal prime ideals is A. Before we move to describe Fmspaces, we first discuss a special class of mquasinormal !rings. Definition 5.1.8. Let A be a local /ring. An embedded prime ideal P is high if for every minimal prime ideal N E Min(A), either N P or NV P = Iex(A). Otherwise, P is low. Call an !ring A a broom ring if for every maximal ideal M every prime ideal in AM is high. The following is immediate from Proposition 2.5.2 and Lemma 5.1.3. Proposition 5.1.9. Let A be a local commutative semiprime /ring with identity and maximal ideal M. The following are equivalent: 1. A is a broom ring. 2. 1r(A) 2. 3. If P Iex(A) is a prime ideal, then rkA(P) 1.
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92 4. If a, b E A are disjoint and P is a proper prime ideal containing a.1 + b.1, then lex(A) P M. 5. If a, b EA are disjoint and a.1 + b.1 lex(A), then for every p E lex(A), there exists z rt, lex(A) such that zp E a.1 + b.1. Proof: (1) # (2) : Since every prime ideal of A is high, we have that for any two distinct minimal prime ideals Qi, Q2 that Qi V Q2 = lex(A). Hence 1r(A) :::; 2 by definition. Conversely, 1r(A) :::; 2 implies that every minimal prime ideal is high and therefore that every prime ideal is high, as desired. (2) # (3) : Since 1r(A) :::; 2, we have that rk(P) :::; 1 for any embedded prime ideal P, by Proposition 2.5.2; and vice versa. (2) (4) : By Proposition 2.5.2, we have that a.1 + b.1 = lex(A). (4) # (5): Follows from Lemma 5.1.3. (4) (3) : Assume that (4) holds. Let P be an embedded prime ideal and assume, by way of contradiction, that rk(P) 2. Let Qi, Q2 P be minimal prime ideals. Let Q1 E Qf \ Q2 and let Q2 E Qt \ Q1. Disjointify by defining Qj = Qj Q1 /\ Q2 for j = 1, 2. Then Q1 .1 Q2 since Q1 rt, Q2 and Q2.1 Qi since Q2 rt, Qi. Then q 1 .1 + q 2 .l P and hence lex(A) P, which contradicts that P is embedded. Example 5.1.10. We now present an example of an /ring which is 3quasinormal but is not a broom ring. Let Ao = A1 = [0, oo) 1R, and A2 = Aa = (1, oo) 1R, where each is inversely ordered. Identifying the copies of 0 in the disjoint union A 1 = (Ao II A 1 )/(0 0 ~ 0 1 ) and letting A 2 be the disjoint union A 2 II A 3 we obtain a root system A= A 1 IIA 2 with the induced ordering which we now describe. That is, r < s if and only if either r, s E Aj for j = 0, 1, 2 or 3 and r
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93 We endow A with a partial binary operation. To begin, note that we define 0o + 0o = 0 1 + 0 1 = 0o + 0 1 = 0 0 ~ 0 1 Let r;, s; E A; be nonzero for j = 0, 1, 2, 3. Let ( ); denote the sum in parentheses as the usual sum of real numbers residing in A;; the mark "" signifies that the sum is undefined. + ro so rt st r2 s2 rs 83 ro (2rolo (ro + solo so (ro + so)o (2so)o rt (2rth St +rt t (r2 + r1)2 (s2 + rt)2 (rs+ ri)s (ss + ri)s s1 (r1+sth (2sth (r2+st)2 (s2 + s1)2 (rs+s1)s (s3 + St)s r2 (rt + r2 2 s1 + r2 2 (2r2)2 (s2 + r2)2 s2 (rt + s2 2 s1 + s2 2 lr2 + s2)2 (2s2)2 rs (rt +r3J3 St + rs s (2rs)s (ss + rs}s 83 (rt+ s3 s St+ 83 3 (r3 + s3)3 (2s3)3 Let Co = Ao, C1 = A1, C2 = A1 II A2, and C 3 = A1 II A 3 Then the minimal prime ideals of F(A, JR) are of the form Q; = { u E F(A, JR) : supp(u) A\ C;} for j = 0, 2, 3. The similarly defined Q 1 is a prime ideal by Proposition 3.3.15. Now, it is evident that L = lex(F(A,JR)) = Q 0 V Q2 V Q 3 and Q2 V Qa = Q1 =IL so we know that 1r(F(A, JR)) = 3. Therefore, Proposition 5.1.9 shows that the ring is not a broom ring. Since L is the maximal ideal of the ring, we have shown that F(A, JR) is 3quasinormal. We present a similar example of an broom ring that is not quasinormal. Example 5.1.11. Let Ao = (0, oo) lR, and A 1 = A 2 = (1, oo) lR, where each is inversely ordered. Let A 1 = A 0 and let A 2 be the disjoint union A 1 II A 2 in order to obtain a root system A= A 1 IIA 2 with the induced ordering which we now describe. That is, r < s if and only if either r, s E A; for j = 0, 1 or 2 and r < s in the inversely ordered real numbers; or if r E Ai, s E A 0 ; or if r E A 2 s E A 0 Explicitly, r II s if r E A1 and s E A2. We endow A with a partial binary operation. Let r;, s; EA; for j = 0, 1, 2. Let ( ); denote the sum in parentheses as the usual sum of real numbers residing in A;; the mark "" signifies that the sum is undefined.
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94 + ro so r1 81 r2 s2 ro (2ro)o so +ro o (r1 + roh s1 + ro 1 (r2 +roh (s2 + roh so ro + so o (2so)o (r1 + soh 81 + so 1 1r2 +soh (s2 + soh r1 ro + r1 1 so+ r1 1 (2r1)1 s1 + r1 1 s1 ro + s1 1 so+ s1 1 (r1 + s1h (2s111 r2 ro +r2 2 so+ r2 2 (2r2)2 (s2 + r2)2 s2 ro + s2 2 so+ s2 2 (r2 + s2)2 {2s2)2 Let Co = .6.o, C 1 = .6. 0 II .6. 1 and C2 = .6.o II .6.2. Then the minimal prime ideals of F(.6.,R) are of the form Q; = {u E F(.6.,R.) : supp(u) .6. \ C;} for j = 1,2; the ideal Q 0 is a prime ideal by Proposition 3.3.15. Now, Qo = lex(F(.6., R)) = Qi V Q2 and so we know that 1r(F(.6., R)) = 2. Therefore, Proposition 5.1.9 shows that the ring is a broom ring. Since Q 0 is not the maximal ideal of the ring, we have shown that F(.6., :R) is not quasinormal. D 5.2 (B, m)Boundary Conditions Definition 5.2.1. Let m be an integer greater than 1 and let {U;}~ 1 be a family of m pairwise disjoint cozerosets of a topologirol space X. The subspace nj=l clx(U;) is rolled an mboundary in X. Let B be a topological property. We say that a space X satisfies the (B, m)boundary condition if every mboundary in X has property B. In [Lal, 3.5], Larson proves that if Xis completely regular, every point of X is a G5point, and C(X) is quasinormal, then X satisfies the (discrete, 2)boundary condition. This result is improved in [La3, 3.5) to say that if X is normal and for every p E /3X \ vX, the t'ideal QP is prime then C(X) is quasinormal if and only if X has the (finite, 2)boundary condition. Here we refine this theorem by removing the restriction on the points of the corona. First, a lemma, extending [La3, 3.1), which we henceforth refer to as "Larson's Lemma": Lemma 5.2.2. Let X be normal and let {g;}~ 1 C(X)+ be a family of pairwise disjoint functions. Define Ym = n;:1 cl X ( coz(g;)). Then E;:1 gf = r\eYm My.
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95 Proof: Each function in E1=1 9f must vanish on Ym, hence E1=1 9f nyEYm My. For the reverse inclusion, we use recursion. The proof of the base case of m = 2 is in [La.3, 3.1). Let f E nyEYm My and define fi(x) = {Jo I\ 1 if x E cl(coz(91)), if X E n;: 2 cl ( coz (9;)). Since X is normal and Ji is defined on a closed set, the function has a continuous extension, 1 1 E C(X). Then f /\11 1 E 9f and 1 1 E n{Mp: p E nf= 2 cl(coz(9;))}. Recursively define a function 1 k to be the continuous extension of fk(x) = {1kl I\ 1 ~f X E cl~z(9k)) 0 If X E nj=k+l cl(coz(9;)). Then lk1 I\ 1lk E 9f and 1k E n{Mp: p E nj=k+l cl(coz(9;))}. In particular, by the base case, we have that 1 m2 E n{Mp : p E ni=mI cl(coz(9;))} = 9~1 + 9~.i m .i But then f ma /\ 1 f m2 E 9m2 implies that f ma /\ 1 E E;=m2 9; and therefore m .i we have that f ma = (f ma /\ 1) (f ma V 1) E E;=m2 9; Thus, by recursion, we m .1 m .i deduce that f 1 E E;= 2 9; Hence f I\ 1 = (f I\ 1f 1 ) + f 1 E E;= 1 9; and therefore, f = (f I\ l)(f V 1) E E1=i 9f, as desired. Let X be normal and let Ym, as above, be given. The set Ym is Cembedded in X by [GJ, 3D). Thus we have a surjective ring homomorphism cp: C(X) C(Ym) given by restriction of functions. The kernel of the homomorphism is m K= {f E C(X): Ym Zx(f)} = n My= L9f, yEYm j=l by Larson's Lemma. Thus by the First Isomorphism Theorem, it follows that m C(Ym) ~ C(X)/K = C(X)f L9f. j=l We utilize the onetoone correspondence between the prime ideals of C(Ym) and prime ideals of C(X) which contain K, the kernel of cp.
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96 Theorem 5.2.3. Let X be normal and m 2 an integer. The ring C(X) ism quasinormal if and only if X satisfies the ( P, m )boundary condition. Proof: Let Ym = ni=l cl(coz(g;)) be an mrboundary. Ym is a Pspace if and only if the prime ideals of C(Ym) are both minimal and maximal by [GJ, 4J]. By the discussion above and Larson's Lemma, this is also equivalent to the condition that the prime ideals of C(X) containing EJ=l gf are maximal. In turn, this is equivalent, by Theorem 5.1.5, to the statement that C(X) is mrquasinormal. By [GJ, 4K.1], we have that countably compact Pspaces must be finite and discrete. In this light, the following is immediate. Corollary 5.2.4. Let X be normal. 1. If X is countably compact, then C ( X) is mquasinormal if and only if X satisfies the (finite, m)boundary condition. 2. If X is locally compact, then C(X) is mquasinormal if and only if X satisfies the ( discrete, m )boundary condition. S. If X is (1compact, then C(X) is mquasinormal if and only if X satisfies the ( countable discrete, m )boundary condition. Example 5.2.5. (Butterfly Spaces) Let X be a noncompact completely regular space and let S {3X \ X be closed. Define m Bm(X, S) = (ll /3X;)/(S1 ~ S2 ~~Sm), j=l where we assume that X; = X, S; = S for all j = 1, ... m and Si ~ S; indicates that corresponding points of Sare identified, when i =I= j. Consider X = Bm(N, {3N \ N). Let g 1 (x) =when x EN; and let g 1 vanish elsewhere on X. Then {g 1 }J!, 1 is a pairwise disjoint set of functions such that we
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97 have Ym = ni= 1 clx(coz(g;)) = (3N \ N, which not a Pspace. Thus Bm(N, (3N \ N) is not an Fmspace by Theorem 5.2.3. Every point of this space has rank less than or equal tom, thus it is an Fm+1space, by Theorem 5.1.5. The minimal prime ideals contained in Mp for p E (3N \ N and j = 1, ... m, are given by: We show that C(Bm(N,/3N\N)) is a broom ring by demonstrating that Qf, f; Q~+Q; for all j 3. Let f E Qf, and represent the function by an mtuple (/1, h, ... f m) where fk = f l.BN 4 for all k = 1, ... m. For some j 3, let 91 = (f;,h,h,.,.,fm) and 92 = (/1 J;,0,0, ... ,0). Then Yi E Q! for i = 1, 2 and/= 91 + 92 E Q~ + Q;. Thus we have shown that 1r(C(Bm(N, (3N \ N))Mp) ::; 2 for all p E Bm(N, (3N \ N). Hence C(Bm(N, (3N \ N)) is a broom ring by Proposition 5.1.9. Let cp: C(Bm(N,/3N \ N)) C(Ym) be the canonical surjection with kernel K = Ei= 1 gf. We want to explicitly describe the root system structure at each point of the space Bm(N, (3N \ N). For each j, the points p E N; are isolated and hence the spectrum at each of these points consists only of the maximal ideal. If p E f3N\N is a Ppoint then Mp E Max(C(Ym)) is minimal. Hence Mp= cp+(Mp) is a maximal ideal of C ( Bm (N, (3N \ N)) and is minimal with respect to containing K Thus, the maximal ideal Mp is ramified and the graph of Spec(C(Bm(N, f3N\ N).M) p has the form: (5.1)
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98 If p E ,BN \ N is an Fpoint which is not a Ppoint, then Mp E Max(C(Ym)) properly contains a unique minimal prime ideal, namely Op. Hence Mp = cp+(Mp) is a maximal ideal of the ring C(Bm(N, ,BN \ N)) and the ideal Op= cp+(Op) is the unique prime ideal of C ( Bm (N, ,BN \ N)) which is minimal with respect to containing K. Thus, Q;+Q; = E~ 1 Qf, =Op~ Mp and the graph ofSpec(C(Bm(N, ,BN\N)M,,) has the form: (5.2) The following is also immediate from Theorem 5.2.3: Corollary 5.2.6. Let X be normal. 1. Let X be a noncompact, locally compact, acompact Fspace and S a closed subspace of the corona. Then B 2 ( X, S) is an Fr space if and only if S is finite. 2. Let X be an Fspace and S a nowhere dense zeroset of X. Let X; = X and let S = S; X; for j = 1, 2. Define Y = (X1 II X2)/(S1 ~ 82), where corresponding points of S are identified. Then Y is an Frspace if and only if S is a Pspace in the subspace topology. Proof: (1) By [GJ, 6.9], the hypotheses on X imply that ,BX\ Xis compact and X is a cozeroset of ,BX. Thus S is compact and there exists f E C (,BX) such that coz(f) = X. Let fi(x) = {f 0 (x) ifx E ,BX 1 if XE X2 h(x) = {fo(x) if XE ,8X2 if XE X1
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99 Then cl(coz(f 1 )) n cl(coz(f 2 )) = S. If B 2 (X, S) is an F2space, then Sis a compact Pspace by Theorem 5.2.3. Thus, Sis finite by [GJ, 4K.l]. Conversely, assume that S is finite. Since X is an Fspace, we know that any 2boundary of B 2 (X, S) must be contained in Sand must therefore be finite. Thus, every 2boundary is a Pspace and we conclude that B2(X, S) is an F2space by Theorem 5.2.3. (2) As above, any 2boundary of Y is contained in S. In particular, S itself is a 2boundary since it is a nowhere dense zeroset of X. Thus, if Y is an F2space, then Sis a Pspace, by Theorem 5.2.3. Conversely, since any subspace of a Pspace is a Pspace by [GJ, 4K.4], we know that every 2boundary of Y is a Pspace. Therefore, Y is an F2space. Next, we show that it is the case that any normal Pspace arises as a 2boundary in an F2space. Proposition 5.2.7. Let Y be a normal Pspace. Then there exists an F2space X containing Y such that in the subspace topology Y = clx(coz(Je)) n clx(coz(J 0 )), where le, lo E C(x)+ are disjoint. Proof: Let Y be a normal Pspace and for each y E Y, let a 11 N 11 be the onepoint compactification of the natural numbers in which ay plays the role of the point at infinity. Define X = (YU (llyEY a 11 N 11 ))/(a 11 ~ y). Let the points of N remain isolated. A base for the Xneighborhoods of y E Y is given by the sets of the form Uy U (Uy'EU 11 Ny1 ), where Uy is a neighborhood of yin Y and Ny, is a neighborhood of ay' in ay, Ny, Define fe on X such that fe(n) = for all even n E Ny and fe(x) = 0 otherwise. Similarly, define a function f 0 on X such that Jo( n) = if n E Ny is odd, and Jo(x) = 0 otherwise. Then fe A J 0 = 0 and Y = clx(coz(Je)) n clx(coz(J 0 )).
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100 Since the only points of X having rank greater than 1 are contained in the subspace Y, we have that all the 2boundaries are subspaces of a Pspace. Thus by [GJ, 4K.4] and Theorem 5.2.3, we have that X is an F2space. It is natural to consider the implications of boundary conditions other than for Pspaces. We now take a look at Fspace boundary conditions. Theorem 5.2.8. Let X be normal. The following are equivalent: 1. X satisfies the (Fspace, n)boundary condition. 2. Let p E /JX. If { P 1 ... P 2 n} MP are pairwise incomparable prime ideals, then E;= 1 P; and E~n+i P; are comparable. 3. There does not exist a pair of noncomparable prime ideals contained in the same maximal ideal such that each has rank greater than or equal to n. That is, the graph of Spec(C(X)) does not contain a copy of n n Proof: The equivalence of (2) and (3) is clear. (1) =} (3) : Let p E /JX be such that MP contains two incomparable prime ideals, P, Q, each of rank greater than or equal to n. Let P 1 ... Pn P and Q 1 ... Qn Q be distinct minimal prime ideals. For k = l, ... n, let Then by convexity and primeness, fk = f1k/\hk E (QtnP:)\((U#k Q;)U(LJ,# P,)) We disjointify by defining gk = /\ifkfi A'J= 1 f;Then gk (/. Qk U Pk and hence, gt Qk n Pk. Thus E:= 1 gt (E;= 1 Qk) n (E;= 1 Pk) P n Q.
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101 IT y = n~=l cl(coz(gk)), then by the prime correspondence discussed before Theorem 5.2.3, and by the construction of the functions {9kH1= 1 the incompa rable prime ideals P and Q of C(X) give rise to the incomparable prime ideals P/'E~= 1 gt,Q/"~= 1 gt MP/E~= 1 gt in C(Y). Therefore, Y is an nboundary which is not an Fspace. (3) ::::} (1) : Let Y = ni=l cl(coz(Jj) be an nboundary in X and assume that Y is not an Fspace. Then there exists p E (3Y such that Wy contains incomparable prime ideals P, Q. Since C(Y) ~ C(X)/ Ei=l ff, we have that M~ contains the two incomparable prime ideals P, Q such that P / Ei=l ff ~ P and Q/ Ej=l ff ~ Q. Thus, since Ei=l ff~ P, we know that rkc(x)(P) n, by the pigeonhole principle and [BKW, 3.4.12). Likewise, rkc(x)(Q) n. Example 5.2.9. Consider the ordinal X = w 2 + 1 in the interval topology. Let E be the subset { >. + 2n : n E w, >. limit ordinal} be the set of even ordinals less than w 2 + 1 and let D be the subset {>. + (2n + 1) : n E w, >. limit ordinal} be the odd ordinals less than w 2 + 1. Define fe(x) = if x = >. + 2n E E, and let fe vanish otherwise on X. Let f 0 ( x) = if x = >. + (2n + 1) E D and let f O vanish otherwise on X. Then Y = clx(coz(Je)) n clx(coz(/ 0 )) = {nw: n E w} U {w 2 } is homeomorphic to the one point compactification of w where w 2 acts as the point at infinity. This 2boundary is not an Fspace. Thus, via analysis similar to the proofs above, the canonical surjection and [GJ, 14G), we know that the spectrum at Mw2 in C(w 2 +1) contains the following subgraph: (5.3)
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102 In fact, if, form~ 2, we partition N into m countably infinite sets {A;}J!= 1 and let f( x) = { if x = A + n for n E A; 3 0 else. Then Y = nJ=l cl(coz(f;)) = {nw : n E w} U {w 2 } is an mboundary in w 2 + 1. Thus each branch in the graph above that emanates from the maximal ideal has at least 2' minimal vertices, by Proposition 2.3.3. The special Fspaces, extremally and basically disconnected spaces, have a different sort of boundary description. Let us review a couple of results. Recall that a completely regular space X is cozerocomplemented if for any cozeroset U X there exists a cozeroset V X such that U n V = 0 and U U V is dense in X. Dually, an /ring A is complemented if for every f E A+ there exists g E A+ such that f I\ g = 0 and / + g is not a zerodivisor. Lemma 5.2.10. Let X be completely regular. 1. The following are equivalent: (a) C(X) is complemented. (b) X is cozerocomplemented. (c) Min(C(X)) is compact. 2. X is a cozerocomplemented F space if and only if X is basically disconnected. 3. X is extremally disconnected if and only if Min(C(X)) a,{ Max(C(X)) is ex tremally disconnected. Proof: (1) The equivalence of (a) and (c) is shown in [M, 1.5). The equivalence of (a) and (b) is from the definitions. (2) See [HVWl, 2.16]. (3) See [M, 2.6] and [M, 2.7].
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103 Let F C(X). Under the hullkernel topology on Spec(C(X)), we define a subspace homeomorphic to Spec( C ( X) / E /EF f .1) by PF= {PE Spec(C(X)): Pis minimal with re.spect to containing Lf.1}. /EF Proposition 5.2.11. Let X be normal. The following are equivalent: 1. X satisfies the (basically disconnected, n)boundary condition. 2. X satisfies the (cozerocomplemented F, n)boundary condition. S. If F = {/;}7= 1 c C(X)+ is a pairwise disjoint family, then PF is compact and there does not exist a pair of noncomparable prime ideals contained in the same maximal ideal such that each has rank greater than or equal to n. Proof: (1) <=> (2): Follows from (2) of Lemma 5.2.10. (2) <=> (3) : The compactne.ss of PF follows by (1) of Lemma 5.2.10 and the prime corre,spondence discussed before Theorem 5.2.3. The final statement of (3) is a result of Theorem 5.2.8. Proposition 5.2.12. Let X be normal. The following are equivalent: 1. X satisfies the (extremally disconnected, n)boundary condition. 2. If F = {J;}J=l C C(X)+ is a pairwise disjoint family, then PF is extremally disconnected and compact and there does not exist a pair of noncomparable prime ideals contained in the same maximal ideal such that each has rank greater than or equal to n. Proof: Follows directly from (3) of Lemma 5.2.10, the prime corre.spondence dis cussed before Theorem 5.2.3 and from Theorem 5.2.8. Next we describe the spectra of C(X) which guarantee the (Fm, n)boundary condition on X.
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104 Definition 5.2.13. Let a EN and let n > 0 be an integer. Then we call thentuple a = (a 1 "2, ... an) a good (ordered) partition of a if the following hold: for all j = 1, ... n, we have that O < a; E Z, a= a1 + a2 ++an, a1 > 1 and a1 "2 Construct an upwarddirected graph ~a corresponding to a good partition a of a containing: nodes P1,P2, ... ,Pn such that P1 < P2 < < Pn, each P; lies above s; = a 1 + 02 terminal vertices and there is a node q such that q > Pi for every j = 1, 2, ... n. As follows: (5.4) Proposition 5.2.14. Let A be a commutative semiprime /ring with identity in which the sum of any m distinct minimal prime ideals is a prime lideal. Then A is mquasinormal if and only if Spect(A) does not contain ~a as a subgraph for any good ordered partition a of m. Proof: Assume that Spect(A) contains ~a as a subgraph for some good ordered partition a = ( a1, .. an.) of m. Let Pk be the prime ideal at the node Pk for each k = 1, ... n. For a fixed k, let Qk = { Qk;} ;~ 1 be a family of distinct minimal prime ideals contained in Pk but not contained in P; for j < k. Then LJ~ 1 Qk is a family of m distinct minimal prime ideals whose sum is the nonmaximal prime ideal Pn.. Thus A is not mquasinonnal. Conversely, if A is not mquasinonnal, then there exists a family of m distinct minimal prime ideals { Qk}r=l such that :E;;=l Qk is not a maximal ideal. Let
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105 P 1 = E;:= 1 Q1r, and let M be the maximal ideal containing it. Then the subgraph of Spec(A) bounded by the vertices corresponding to the prime ideals in the set { Pi, M} U { Q1r,}~ 1 is of the form .6. 0 for some a. Example 5.2.15. The good ordered partitions of 4 are (4), (3, 1), (2, 2) and (2, 1, 1). If Xis normal, then it satisfies the (P, 4)boundary condition if any only ifSpec(C(X)) does not contain any subgraphs of the following forms: (5.5) This follows from Theorem 5.1.5 and Proposition 5.2.14. Obtain the graph Ll~ by appending a graph having at least k 2 minimal vertices to each terminal vertex of the graph .6. 0 For instance: (5.6) The following is a consequence of Proposition 5.2.14 and the method of proof of Theorem 5.2.3. Corollary 5.2.16. Let X be nonnal. Then the following are e.quivalent: 1. X satisfies the (Fm, k)boundary condition.
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106 2. Spec(C(X)) does not wntain a subgraph of the form~~ for any good ordered partition a of m. 9. Let a family P Spec(C(X)) of mk nonwmparable prime ideals and a good partition a= (ai, a 2 , an) of mk be given. Partition Pinto pairwise disjoint sets {P;}ff 1 where P; contains a; elements of P. Then at least two of the prime ideals given by E P; are wmparable. Proof: The equivalence of (2) and (3) is clear. (1) => (2) : Assume that X contains a kboundary Y = f\~ 1 cl(coz(f,)) which is not an Fmspace. Then by Proposition 5.2.14, Spec(C(Y)) contains a subgraph of the form ~ 0 below some maximal ideal Mt for some good partition a= (a 1 ... an) of m and some p E f3Y. For each i = 1, ... n, let Pi be the prime ideal at the node Pi E ~ 0 and let {Qi;} i 1 be a family of distinct minimal prime ideals contained in but not contained in P, for l < i. Then, by the prime correspondence discussed before Theorem 5.2.3, there exist, for each i, prime ideals ~' Qi; in C(X), such that ~ I'i/ Ef= 1 f/and Qi; '.:::'. Qi;/ Ef= 1 J,1for each j. Then by [BKW, 3.4.12), rk(Qi;) k for all i,j. Therefore, the spectrum below M~ contains a copy of~~ (2) => (1) : Assume that the spectrum below M~ contains a copy of~! for some good partition a= (ai, ... an) of m. For each i = 1, ... n, let~ be the prime ideal at the node Pi E ~! and let { Q;;} i 1 be a family of distinct nonmaximal as well as nonminimal incomparable prime ideals contained in~' but not contained in P,, for l < i. For each pair i,j such that 1 i n and 1 j ai, let {Qi;lf= 1 be a family of distinct minimal prime ideals contained in Qi; but not contained in Qr., if r < i or if r = i and s /= j. Let !str E Q:; \ (U lfr Q~J Then, by convexity, we have that for all l
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107 E:=1 g; C Qi; for all pairs i,j. Then Y = n;=l cl(coz(gr)) is not an Fmspace since Mt contains b.. 0 as a subgraph, by our choice of the functions. 5.3 {jX, mQuasinormal and SV Conditions First we give equivalent conditions under which one may expect that f3X is an Fmspace for some integer m. This improves [La2,4.3] and extends [La2, 3.3]. Lemma 5.3.1. Let X be normal and let Ym = {x EX: rkx(x) > m 1}. 1. Let y E f3X \ X. If rkpx(y) > m 1, then y E clpx(Ym) 2. If Ym is compact, then rkpx(Y) :Sm 1 for every y E f3X \ X. Proof: (1) Let U be a f3Xneighborhood of y. Then there exists a closed f3X neighborhood V of y such that V U and V n X is closed in X. Since V n X is C*embedded in X, we have that y E f3(V n X) = clpx(V n X) V. Now, by [La2, 1.6], we have that V n Ym =J 0 since rkpx(Y) > m 1. (2) Let y E f3X \ X. If rkpx(Y) > m 1, then by (1), y E clpx(Ym) X, which is a contradiction. Hence, rkpx (Y) :S m 1. Using the lemma above and Theorem 5.2.3, we obtain: Theorem 5.3.2. Let X be normal and let m 2 be an integer. The following are equivalent: 1. C(X) is mquasinormal and rkx(MP) :Sm 1 for every p E f3X \ X. 2. X satisfies the (finite, m)boundary condition. 9. X contains only finitely many points of rank greater than m 1. 4f3X contains only finitely many points of rank greater than m 1. 5. C(f3X) is mquasinormal.
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108 Proof: The implications (2) =} (3) =} (4) =} (5) have the exact same proofs as in [La2, 4.3], using Lemma 5.3.1. (5) =} (2) : Let { coz(fi) }~ 1 be a pairwise disjoint set of cozerosets of X, and for each i, let ff be an extension of Ji to (3X so that { coz(ff) }~ 1 is a pairwise disjoint family. Then n:, 1 clpx(coz(ff)) is finite by Theorem 5.2.3. Since n:, 1 clx(coz(fi)) is contained in this finite set, it too is finite. Thus, (2) holds by Theorem 5.2.3. (1) =} (5) : Follows from Theorem 5.1.5. (3) =} (1) : Since (3) implies (5), we have that C(X) is mquasinormal. Condition (3) also implies that rkpx(p) = 1 for all p E (3X \ X, by Lemma 5.3.1. Recall that an /ring A is called an SVring if A/ P is a valuation ring for every prime ideal P. The following arises when in pursuit of conditions which imply that C(X) is both mquasinormal and SV. Theorem 5.3.3. LetX be normal andm 2. IJC(X) is mquasinormal and every maximal ideal has finite rank, then X satisfies the (finite, m)boundary condition. Proof: Assune that C(X) is mquasinormal, that every maximal ideal has finite rank and that {U; : 1 :S j :S m} is a family of pairwise disjoint cozerosets such that the set W = nf= 1 clx(U;) is infinite. Then there exists a copy of Nin W. Denote this copy by Y = {x; : j EN}. Then Y is not closed in (3X, so there exists p E (3X such that p E clpx(Y) \ Y. Let {U_; : 1 :S j :S m} be cozerosets of (3X such that u; n X = U; for each j Then p rJ_ u; for all j = 1, 2, ... m and p E nf= 1 cl 13 xu;. For each i, Xi E W and hence rkx(xi) m. Let Pii, P 2 i, ... Pmi be distinct minimal prime ideals of C(X) such that we have P;i Mxi for all j = 1, 2, ... m and U; (/. ukhcoz(Pki)Let p E (3X and rkc(X)(MP) = n < 00. Then as shown in the proof of [L,4.2], there exists a minimal prime ideal Q MP such that for every Z E Z(Q) such that Zn Y is infinite.
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109 For every i E C(X) define Jj = { xi E Y : i E P;i} for j = 1, 2, ... m. Let :F be the zultrafilter on Y containing the zfilter {Zn Y : Z E Z(Q)}. Define for j = 1, 2, ... m, R; = {i E C(X) : J1 E :F}. As shown in [L,4.2], each R; is a prime ideal of C(X). First we show that the R; are noncomparable ideals. For each Xi E Y, let Ui be a neighborhood of Xi such that the collection of these neighborhoods is pairwise disjoint. For each i E N and for each 1 5 l, k 5 m, let iu:i E C(X) such that i1ki E Pii \ Pki, i(X \ Ui) = 0, and O 5 ilki 5 ~. Then let i1k = E: 1 i1ki Finally, i1k E R1 \ Rk and ikl E Rk \ R1 for all 1 5 l, k 5 m, l =I= k. Let R'; R; be a minimal prime ideal for each j = 1, 2, ... m. Larson demon strates that R; MP for all j. We will show that E;, 1 R'; is neither the maximal ideal MP nor all of C(X). Define h E C(clx Y) by h(xi) = for i = 1, 2, .. and let h(x) = 0 otherwise. Then since X is normal and clx Y is closed, h extends to a function h E C(X) such that h E MP. We show that h 'IE;, 1 R;. For each j = 1, 2, ... m, let!; ER;. Then Jji E :F and hence ni== 1 J1i E :F and (E;, 1 !;)(xi) = 0 for every Xi E ni== 1 Jt If h = E;, 1 i; then since h(xi) =/= 0 for all Xi E Y, 0 = ni== 1 Jji E :F, a contradiction. Thus h E MP\ E; 1 R; and therefore, E; 1 If; E1=i R; MP and C(X) is not mquasinormal, a contradiction. Thus for any m pairwise disjoint cozerosets {U;: 1 5 j 5 m} the set ni= 1 clxU; is finite.
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[AF] [AM] [BKW] [BP] [Ce] REFERENCES M Anderson and T. Feil, LatticeOrdered Groups, Reidel (Kluwer), Dor drecht, 1988. M. Atiyah and I. Macdonald, Introduction to Commutative Alge bra, AddisonWesley Publishing Company, ReadingLondonAmsterdam Sydney, 1969. A. Bigard, K. Keimel, and S. Wolfenstein, Groupes et Anneaux Reticules, Springer Lecture Notes in Mathematics (608), Springer Verlag, Berlin HeidelbergNew York, 1977. G. Birkhoff and R. S. Pierce, Latticeordered rings, Anais. Acad. Brasil. Cien. (29), 1956, pp. 41 69. E. Cech, On bicompact spaces, Annals of Math. (38), 1937, pp. 823 844. [ChDil] G. Cherlin and M. Dickmann, Real closed rings, II. Model theory, Ann. Pure Appl. Logic (25), 1983, pp. 213 231. [ChDi2] G. Cherlin and M. Dickmann, Real closed rings, I. Residue rings of rings of continuous functions, Fund. Math. (126), 1986, pp. 147 183. [Cl] [C2] [C] [CD] [CHH] [CM] [CMc] [D] P. Conrad, Generalized semigroup rings, J. Indian Math. Soc. (21), 1957, pp. 7395. P. Conrad, Generalized semigroup rings, II, Portugaliae Math. (18), 1958, pp. 33 53. P. Conrad, Latticeordered Groups, Tulane Lecture Notes, Tulane Univer sity, 1970. P. Conrad and J. Dauns, An embedding theorem for latticeordered fields, Pacific J. Math. (30), 1969, pp. 385 398. P. Conrad, J. Harvey, and W. Holland, The Hahn embedding theorem for latticeordered groups, Trans. Amer. Math. Soc. (108), 1963, pp. 143 169. P. Conrad and J. Martinez, Complemented latticeordered groups, Indag. Mathern., N. S. (1), 1990, pp. 281 298. P. Conrad and P. McCarthy, The Structure of /Algebras, Math. Nachr., (58), 1973, pp. 169191. M. Darnel, Theory of LatticeOrdered Groups, Marcel Dekker, New York Basel, 1995. 110
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[DHH] [vD] [E] [Fu] [GK] [GJ] [G] [Ha] [HaM] [HIJo] [HJ] [HJo] [HLM] 111 F. Dashiell, A. Hager, and M. Henriksen, OrderCauchy completions of rings and vector lattices of continuous functions, Canad. J. Math (32), 1980, pp. 657685. E. van Douwen, Remote Points, Dissertationes Math. (Rozprawy Mat.) (188) 1981, pp. 145. R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989. L. Fuchs, Partially Ordered Algebraic Systems, AddisonWesley, Reading Palo AltoLondon, 1963. I. Gelfand and A. Kolmogoroff, On rings of countinuou.s functions on topological spaces, Dokl. Akad. Nauk SSSR (22), 1939, pp. 11 15. L. Gillman and M. Jerison, Rings of Continuous Functions, Graduate Texts in Mathematics, Springer Verlag (43), BerlinHeidelbergNew York, 1976. R. Gilmer, Multiplicative Ideal Theory, Pure and Applied Mathematics (12), Marcel Dekker, New YorkBasel 1972. A. Hager, Minimal covers of topological spaces, Ann. NY Acad. Sci, Papers on Gen. Topol. and Rel. Cat. Th. and Top. Alg. (552), 1989 pp. 44 59. A. Hager and J. Martinez, Fractiondense algebras and spaces, Canad. J. Math (45), 1993, pp. 977 996. M. Henriksen, J. R. Isbell, and D. G. Johnson, Residue class fields of latticeordered algebras, Fund. Math. (50), 1961, pp. 107 117. M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. (82), 1965, pp. 110 130. M. Henriksen and D. G. Johnson, On the structure of a class of archimedean lattice ordered algebras, Fund. Math. (50), 1961, pp. 73 93. M Henriksen, S. Larson, and J Martinez, The Intermediate Value The orem for polynomials over latticeordered rings of functions, Papers on General ToEology and its Applications (Amsterdam, 1994), Ann. NY Acad. Sci. (788), 1996, pp 108123. [HLMW] M. Henriksen, S. Larson, J. Martinez, and R. G. Woods, Latticeordered algebras that are subdirect products of valuation domains, Trans. Amer. Math. Soc. (345) 1994, pp. 195 221. [HVWl] M. Henriksen, J. Vermeer, and R. Woods, Quasi Fcovers of Tychonoff spaces, Trans Amer. Math. Soc. (303), 1987, pp. 779803. [HVW2] M. Henriksen, J. Vermeer, and R. Woods, Wallman covers of compact spaces Dissertationes Mat. (Rozprawy Mat.) (280), 1989, pp. 1 31. [Hu] C. Huijsmans, Some analogies between commutative rings, Riesz spaces, and distributive lattices with smallest element, Indag. Math. (36), 1974, pp. 132 147.
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(HudP] 112 C. Huijsmans and B. de Pagter, Maximal dideals in a Riesz space, Canad. J. Math. (35), 1983, pp. 1010 1029. (HudPI] C. Huijsmans and B. de Pagter, On zideals and dideals in Riesz spaces, I, Nederl. Akad. Wetensch. Indag. Math. (42), 1980, pp. 183 195. (Kl] (K2] [K3] (Lal] [La4] [La2] (La3] (Le] (LZ] (M] (MW] (PW] (Sh] [St] [VJ (Wa] (Wo] C. Kohls, Ideals in rings of continuous functions, Fund. Math (45), 1957, pp. 28 50. C. Kohls, Prime ideals in rings of continuous functions, Illinois J. Math. (2), 1958, pp. 505536. C. Kohls, Prime ideals in rings of continuous functions, II, Duke Math. J. (25), 1958, pp. 447 458. S. Larson, A characterization of !rings in which the sum of semiprime ideals is semiprime and its consequences, Comm. Alg. (23), 1995, pp. 5461 5481. S. Larson, Convexity Conditions on frings Can. J. Math. (38), 1986, pp. 4864. S. Larson, !Rings in which every maximal ideal contains finitely many minimal prime ideals, Comm. Alg. (25), 1997, pp. 3859 3888. S. Larson, Quasinormal !rings, Ordered Algebraic Structures, Kluwer Academic Publishers, DordrechtBostonLondon, 1997, pp. 261 276. A. LeDonne, On prime ideals of C{X), Rend. Sem. Mat. Univ. Padova (58), 1977, pp. 207 214. W. A. J. Luxemburg and A. Zaanen, Riesz Spaces, I, North Holland, AmsterdamLondon 1971. J. Martinez, On commutative rings which are strongly Prof er, Comm. Alg. (22), 1994, pp. 347~3488. J. Martinez and S. Woodward, Bezout and Prii.fer !rings, Comm. Alg. (20), 1992, pp. 2975 2989. J. Porter and R. Woods, Extensions and Absolutes of Hausdorff Spaces, Springer Verlag, New YorkBerlin 1988. E. Shchepin, Real functions and nearnormal spaces, Soviet Mathematics (13), 1976, pp. 820830. M. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. (41), 1937, pp. 375481. J. Vermeer, The smallest basically disconnected preimage of a space, Topol. and its Appl. (17), 1984, pp. 217 232. R. Walker, The StoneCech Compactification, Ergebnisse der Mathematik und Ihrer Grenzgebiete, Springer Verlag, BerlinHeidelbergNew York, 1974. S. Woodward, On !rings which are rich in idempotents, Doctoral Dis sertation, University of Florida, 1992.
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BIOGRAPIDCAL SKETCH Chawne Monique Kimber was born in Frankfort, Kentucky, on January 12, 1971. She was raised in Tallahassee, Florida, where she graduated from Leon High School in 1988. Chawne received the Bachelor of Science in Mathematics from the University of Florida in 1992, and the Master of Science in Mathematics from the University of North Carolina at Chapel Hill in 1995. 113
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of ilosop <:., v,~lltllt4 Jorge arti ez Chairman Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. @!9tL Associate Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable s tandards of scholarly presentation and is fully adequate in scope and quality, as a dissertation for the degree of Doctor of Philosophy. AJ~ov Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ~Ir th" c,,.h'~ Scott McCullough Associate Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ~{?,~~ Mildred HillLubin Associate Professor of English This dissertation was submitted to the Graduate Faculty of the Department of Mathematics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. May 1999 Dean, Graduate School
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LD 1780 199 UNIVERS I T Y OF FLORIDA 111 \I \\\I\ \ \\\\ \ \\\ \ \\\ \ \\\ \ \ \ \\ \ \\\ \\ \\\ \ \ \I \ \\\ \ \\\\\ \ \\ \ \\\\ \ 3 1262 08554 8427

