On commutative f-rings which are rich in idempotents

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On commutative f-rings which are rich in idempotents
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Full Text











ON COMMUTATIVE f-RINGS WHICH
ARE RICH IN IDEMPOTENTS












By

SCOTT DAVID WOODWARD


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


1992


'.1 ^ ^ r I I










ACKNOWLEDGEMENTS

I would like to express my deep gratitude to my advisor, Dr. Jorge Martinez.

His guidance, perseverance and patience have made this work possible. His concern

for his students and his perspective of the human condition have been a personal

inspiration.

I would like to thank the members of my committee, Drs. Krishna Alladi, Roy

Bolduc, Doug Cenzer and Chris Stark, for their time, interest and input. In addition

I would like thank Dr. Neil White, for his consistent support, and Dr. Robert Long,

for inspiring me to follow my bliss.

My thanks go to the Department of Mathematics for its support during my years

as a graduate student. In particular, I would like to thank the department secretaries

for keeping me on top of things and making my time here pleasant, and Randy Fischer

for helping me with my computer phobia.

Special thanks go to my family; to my parents for encouraging me to be who I

am; to my grandmother, the family's first mathematician; to my children for making

what I do matter.

Finally, a very special thank you to Rita, my wife and best friend. Without her

belief in me, none of this would have happened.

This work is lovingly dedicated to the memory of my father.
















TABLE OF CONTENTS


ACKNOWLEDGEMENTS ............................

ABSTRACT .. .. ... .. .. .. .. ... .. .. .. .. .. .. .


CHAPTERS


1 INTRODUCTION ...............................


Lattice Ordered Groups and the Yosida Space .........
Commutative Semi-Prime f-Rings and the Maximal Spectrum
Tychonoff Spaces and Semi-Prime f-Rings ...........
Boolean Duality ..........................


2 LOCAL-GLOBAL f-RINGS ..........................

2.1 Introduction . . . .
2.2 The Specker Subring of a Semi-Prime f-Ring . .
2.3 A Characterization of Local-Global f-Rings . .

3 QUASI-SPECKER f-RINGS AND f-RINGS OF SPECKER TYPE .

3.1 Introduction . . . .
3.2 Quasi-Specker f-Rings and f-Rings of Specker-Type . .
3.3 Quasi-Specker Spaces and Specker Spaces . .
3.4 The Absolute of a Hausdorff Space. . ...
3.5 Specker Spaces and Absolutes ......................

4 CONCLUSION .................................


REFERENCES ...................................

BIOGRAPHICAL SKETCH.................................


1
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14

19

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28

44













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




ON COMMUTATIVE f-RINGS WHICH
ARE RICH IN IDEMPOTENTS

By

SCOTT DAVID WOODWARD

August 1992


Chairman: Dr. Jorge Martinez
Major Department: Mathematics

In this dissertation we compare algebraic properties of commutative semi-prime f-

rings A having the bounded inversion property with topological properties of Max(A),

the compact Hausdorff space of maximal ideals of A with the hull-kernel topology.

An f-ring A is local-global if for every primitive polynomial f(t) E A[t], there is

an a E A such that f(a) is a multiplicative unit. We prove that if A is a commutative

semi-prime f-ring with identity and bounded inversion, then each of the following

implies the next.


1. A is local-global.

2. For each primitive a bt2 E A[t] with 0 < a, b, there exists a c E A such that

a bc2 is a unit.

3. Max(A) is zero-dimensional.











4. Max(A) Max(S(A)), where S(A) is a subalgebra of A generated by the

idempotents of A.


Furthermore, the last three conditions are equivalent and if A has a strong unit,

then all the conditions are equivalent. As a corollary, we show that for X a compact

Hausdorff space that C(X) is local-global if and only if X is zero-dimensional. As

a special case we show that if A is a Bezout ring or X is a quasi-F space then the

assumption of a strong unit or compactness can be dropped.

We then take a closer look at the containment of S(A) in A. Specker-type and

quasi-specker rings are defined. It is shown that A is a specker-type ring if and only

if A is an f-subring of Q(S(A)), the complete ring of quotients of S(A), if and only if

Q(A) = S(A)L, the lateral completion of S(A).
We then define specker spaces and quasi-specker spaces. It is shown that a space X

is a specker (quasi-specker) space if and only if #X, the Stone-Cech compactification,

is a specker (quasi-specker) space. Finally we show that if X is a quasi-specker space

and EX, the absolute of X, is a specker space, then X is a specker space. The

converse obtains when X is compact and has a countable 7r-base.
















CHAPTER 1
INTRODUCTION

This dissertation is largely an exploration of relationships that exist between

certain algebraic structures and topological spaces that can be associated with one

another in a natural and sometimes functorial manner.

The historical and methodological basis for this comparison goes back to the

pioneering work of Marshall Stone, in particular his paper "Applications of the Theory

of Boolean Rings to General Topology" [29] where he shows that the category of

Boolean algebras with lattice homomorphisms is dual to the category of compact

zero-dimensional Hausdorff spaces with continuous maps. This idea of considering

certain algebraic substructures to be the points of a topological space pervades much

of what follows.

1.1 Lattice Ordered Groups and the Yosida Space

A lattice-ordered group, denoted -group is a group (G, +, 0) together with a par-

tial ordering < on G such that (G, <) is a lattice satisfying a compatibility condition

between the ordering and the group operation:


If a < b then a + c < b + c and c + a < c + b

We will explicitly define what a lattice is in Section 1.4. Although in the general

theory of lattice-ordered groups there is no assumption or necessity that the group

operation be commutative, for the purposes of this dissertation we will assume that

all groups are commutative. A lattice-ordered group which is a vector space over the










reals such that scalar multiplication by positive real numbers preserves the order is

called a vector lattice.

For a, b E G we will denote the least upper bound (join) and greatest lower bound

(meet) of a and b by a V b and a A b respectively. For infinite or arbitrary subsets
of G we will denote respectively the least upper bound and greatest lower bound of

these sets when they exist, by V g\ and A gx when they are indexed and V S and A S

otherwise. G+ will denote the set of all g E G with 0 < g.

Lattice-ordered groups have the following properties. A much more complete list

of the properties of e-groups as well as proofs of these results can be found in Chapter

1 of "Lattice-Ordered Groups" [2].

Let G be an -group, a, 6, c E G, then

1. a+ (bV c) = (a+b)V (a +c) and dually.

2. -(a V b) = (-a) A (-b) and dually.

3. a+ b = (aVb)+(aAb).

4. (G, A, V) is a distributive lattice, that is aA (b6V c) = (a A b) V (a A c) and dually.

5. (G, +) is a torsion free group.

Lattice-ordered groups also have the Riesz Interpolation Property;

6. If hi,..., h, E G+ and if 0 < g < h, +* + h, then there exists g,..., g E G+,

with 0 < gi < hi for 1 < i
For a, b E G we say that a is disjoint to b if a A b = 0. For g E G the positive part

of g is g+ = g V 0 and the negative part of g is g- = (-g) V 0. Then g+ A g- = 0 so

that we can write any element uniquely as the difference of disjoint elements, namely










g = g+ g-. The absolute value of g is lgI = g+ + g-. For (abelian) i-groups we
have the triangle inequality;

7. a + bl < al + Ibl.

From (3) above we have that;

8. If aAb=Othen a+b=aVb.

In the case that an infinite join or meet exists, we have the following infinite
versions of (1), (2) and (4);

9. g + (V hA) = V(g + hx) and dually.

10. -(V hA) = A(-h>) and dually.

11. g A (V hA) = V(g A hA) and dually.

We now turn our attention to the subgroups of an i-group.

Definition 1.1.1 If G is an i-group, a subgroup H of G is called an i-subgroup if H is

a sublattice of G; that is, H is closed under finite meets and joins. An i-subgroup H
is said to be convex if whenever hi < g < h2 with hi, h2 E H, then g E H. A normal
convex i-subgroup is called an i-ideal. If G and H are i-groups, a map q : G --+ H is
called an i-homomorphism if it preserves both the group and lattice structure.

Since we are dealing strictly with abelian i-groups, normality of subgroups is
not an issue. From now on we will not distinguish between convex i-subgroups and
i-ideals. We have the following theorem relating i-homomorphisms and i-ideals.

Theorem 1.1.1 (1.2.1 [2]) Let G and H be i-groups and let q : G -> H be an &-

homomorphism from G onto H.










1. Ker4, the kernel of 0, is an f-ideal.

2. If N is an f-ideal, then there is an ordering on G/N such that G/N is an f-group

and the canonical group homomorphism 0 : G -+ G/N is an f-homomorphism.

3. G/Ker4 is f-isomorphic to H.

The ordering on GIN from (2) is induced by the ordering on G by defining

g + N < h + N if there is a k E N such that g < h + k. That this is a lattice ordering

with (g + N) V h + N = (g V h) + N and dually, is the content of Theorem 1.2.1 [2].

Let C(G) denote the set of e-ideals of G. For A, B E C(G), let A A B = A n B

and A V B = the subgroup generated by A and B. By convexity and the Riesz

Interpolation Property, these operations make C(G) a lattice.

Definition 1.1.2 A lattice A is said to be complete if for every subset S of A, both

V S and A S exist in A. A is called brouwerian if for a, b\ E A, aA (V b\) = V(a Ab\).

We are now ready to state the following theorem which is a result from 1942 due

to G. Birkhoff [6].

Theorem 1.1.2 C(G) is a complete brouwerian sublattice of the lattice of all subgroups

of G.

There are certain types of f-ideals which are distinguishable in C(G) that will be

of particular interest.

Definition .1.13 Let g E G. C E C(G) is said to be a value of g if C is maximal with

respect to not containing g. C is said to be prime if G/C is totally ordered.

We have the following theorems which allow us to distinguish values and prime

f-ideals in C(G). See Chapter 2 [5].










Theorem 1.1.3 Let C E C(G). Then C c n{B E C(G) : C C B} if and only if there
is a g E G such that C is a value of g.

An application of Zorn's Lemma establishes the following theorem.

Theorem 1.1.4 Let B E C(G) with g B. Then there is a value C of g with B C C.

Definition 1.1.4 For a value C, C* = f{B E C(G) : B C} is called the cover ofC.

The following provides several useful characterizations of prime f-ideals.

Theorem 1.1.5 (Theorem 2.4.1 [5]) Let P E C(G). The following are equivalent.

1. P is prime.

2. If A, B E C(G) and An B = P then A = P or B = P.

3. {C E C(G) : P C C} is totally ordered.

4. For a, b G, ifa A b P then aEP or b E P.

5. For a, bE G, if aAb = 0 then a E P or bE P.

It is clear then that a value is prime, and from (4) above it appears that prime f-

ideals behave rather like prime ring ideals. We will eventually see in certain cases just

how far this likeness goes. For now we have the following correspondence theorem.

Theorem 1.1.6 (Proposition 2.4.7 [5]) Let G be an f-group, H an f-subgroup of G.

Then the map P i- P n H is a one-to-one correspondence between the prime f-ideals

of G not containing H and the proper prime f-ideals of H.










For a value V, V* is an e-group having V as an e-subgroup. Since V is a prime

e-ideal of V*, by the correspondence theorem we get that the quotient group V*/V

is a totally ordered e-group with no proper t-ideals. The following will characterize

a particular large class of totally ordered groups.


Definition 1.1.5 An f-group G is said to be archimedean if na < b for all n E N

implies that a < 0.


This is equivalent to, if 0 < a,b E G, then there is an n E N such that

na : b. Therefore in the totally ordered case this becomes the familiar definition of

archimedeaneity.

The following theorem is due to O. H6lder [21]. It is from a paper published in

1901, and the proof is a generalization of the classical construction of the reals from

the rationals using the cut completion.


Theorem 1.1.7 Let G be a totally ordered group. The following are equivalent.

1. G is archimedean.

2. G is f-isomorphic to an additive subgroup of R.

3. G has no proper e-ideals.


We now turn our attention to one of the topological spaces which can be associated

with an -group, namely the Yosida space of an archimedean -group.

Let g E G and denote by G(g) the -ideal generated in G by g. That is G(g) =

{a e G : \al < nng\ for some n E N}. An element 0 < u E G is called a unit if for

any a E G, if a A u = 0 then a = 0; u is called a strong unit if G(u) = G. Note that

a strong unit is a unit.










Suppose now that G is an archimedean f-group with unit u. Let

Yos(G, u) = {V E C(G) : V is a value of u}

The sets of the form Ua = {V E Yos(G,u) : a V}, for all a E G, are a base for

the open sets of a topology on Yos(G, u). We will call Yos(G, u), with this topology

the Yosida space of G with respect to u. This is in fact the hull-kernel topology on

Yos(G, u). We have the following theorem.

Theorem 1.1.8 (Corollary 10.2.5 [5]) Let G be an archimedean I-group with unit u.

Then Yos(G, u) is a compact Hausdorff space.

For a topological space X, denote by D(X) the set of all continuous functions
with values in the extended reals that are real valued on a dense subset of X. That

is,

D(X) = {f : X --+ RU {oo} : f is continuous and f-1(R) is dense in X}

It should be pointed out here that although D(X) inherits a lattice ordering from the

ordering on R U {oo}, it is not generally true that D(X) is a group. The problem

is that one can define f + g by (f + g)(x) = f(x) + g(x) on f-'(R) n g-'(R) and
the domain of definition is a dense subset of X, but there is no guarantee that f + g

so defined can then be extended to all of X. If such an extension does exist, by the

density of f-'(R) f g-1(R) it is unique and in this case we define f + g to be this

unique extension. By an e-group in D(X) we mean a sublattice that is a subgroup

with the group operation as defined above. The following is the Yosida Embedding
Theorem for archimedean f-groups.

Theorem 1.1.9 Let G be an archimedean f-group with unit u > 0. Then there exists i-

embedding q$: G -- D(Yos(G, u)) such that G(G) is an f-group and G is e-isomorphic










to f(G). Moreover, can be taken so that (u) = 1, the constant function 1, and if

V, W E Yos(G, u) then there is a g E G such that (g)(V) # (g)(W).

Theorem 1.1.9 originates in a 1942 paper by K. Yosida [34]. Hager and Robertson

[19] show that the separation of points uniquely determines the Yosida space of an
archimedean -group.
For later results we will need a precise description of the embedding 4 whose

existence is guaranteed by Theorem 1.1.9.

Let V E Yos(G, u). Since G is archimedean, by H6lder's theorem, V*/V is f-
isomorphic to an additive subgroup of R. Let Oy be such an -isomorphism. Then

4: G --+ D(Yos(G, u)) is defined b, for g E G,

Ov(g + V) if g EV*
O(g)(V)= oo if g V* and g + V > V
-oo if g V* and g + V < V


1.2 Commutative Semi-Prime f-Rings and the Maximal Spectrum

For the purposes of this dissertation we will consider only commutative rings with

identity. For more on f-rings, see chapter 9 in "Groupes et Anneaux R6ticul6s" [5].

Definition 1.2.1 A lattice-ordered ring (-ring) is a ring (A, +, -, 0, 1) together with

an ordering < on the elements of A such that the following hold:

1. (A, <, +, 0) is an -group.

2. For every a, b, c A, if a < b and 0 < c then ac < be and ca < cb.

Definition 1.2.2 An f-ring is said to be an f-ring if A is order isomorphic to a sub-

direct product of totally ordered rings.










This is equivalent to,

3. For every a, b, cA with 0 c, ifaAb=0 then acAb= 0 = ca A b.

By an f-algebra we mean an f-ring that is also a vector lattice, and by an f-subring

we mean a subring that is also a sublattice. The following theorem lists some of the

properties of f-rings.

Theorem 1.2.1 (Theorem 9.1.10 [5]) Let A be an f-ring and let a, b, c A. Then,

1. If c 0, then c(a A b) = ca A cb and dually.

2. If a A b = 0, then ab = 0.

3. a2 > 0.

4. If a > 0 and ab > 0 then b > 0.

5. For a, b >0 ab = (a A b)(a V b).

If we are not careful, confusion can result in the context of lattice-ordered rings,

due to the overlapping of terminology from ring theory and the theory of lattice-

ordered groups. In an attempt to keep these ideas distinct and be consistent with

acceptable usage, we repeat some old definitions and state some new ones. By an

-ideal of A we mean a convex t-subgroup of the -group (A, +, <) and by an ideal

of A we mean an ideal of the ring (A, +, -). An ideal P is a prime ideal if ab E P

implies a E P or b E P. An -ideal P is a prime 1-ideal if a A b E P implies a E P

or b E P. An element u E A will be called a multiplicative unit if it is an unit in the

ring (A, +, -), an order unit or strong order unit if it is such in the G-group (A, +, <).

The following example is meant to help make these distinctions clear.

Example Let A be the set of all real sequences. If we define addition and multi-

plication pointwise, then A becomes a commutative ring. If we also order pointwise,










that is {a,} 2 {bn} if a, > b, for all n E N, then A is an f-ring. In fact if we also

define real multiplication pointwise, then A is an f-algebra.

Let B be the set of bounded rational sequences. With operations and ordering

defined as for A, B is a commutative f-ring. B is an f-subring of A, and B is not an

f-algebra. Neither is B an ideal nor an e-ideal of A.

Consider the sequences a = {1,1/2,1/3, } and b = {1,, 1,1, }. In B, a is a

weak order unit, but not a strong order unit or a multiplicative unit. In A, a is a

weak order unit and a multiplicative unit, but not a strong order unit. In fact, A has

no strong order units. As an element of B, b is a strong (hence weak) order unit and

a multipicative unit. In A, b is a weak order unit and a multiplicative unit.

The following important concept is due to M. Henriksen et al.[20].

Definition 1.2.3 An f-ring A is said to have the bounded inversion property if every

1 < a E A is a multiplicative unit.

Denote by Spec(A), Min(A), Max(A) the set of prime ideals, minimal prime

ideals and maximal ideals of A respectively. These become topological spaces when

endowed with their respective hull-kernel topologies. In particular, the basic open

sets of Spec(A) are of the form, for a E A, Ma = {P E Spec(A) : a < P}. A ring A

is said to be semi-prime if it has no nonzero nilpotent elements. This is equivalent

to requiring the intersection of prime ideals to be zero. The following results are well

known and will be used extensively.

Lemma 1.2.1 (Theorem 9.3.1 [5]) Let A be a semi-prime f-ring and let a, b A.

Then ab = O if and only if a A b = 0.


Lemma 1.2.2 (Theorem 9.3.1 [5]) Let A be a semi-prime f-ring. Then P is a minimal

prime ideal if and only if P is a minimal prime e-ideal.










The following two results can be found in Henrikson et al.[20] where the notion
of bounded inversion is introduced.

Lemma 1.2.3 Let A be a semi-prime f-ring. Then A has the bounded inversion prop-

erty if and only if every maximal ideal is an -ideal.

Lemma 1.2.4 Let A be a semi-prime f-ring with identity and bounded inversion. Then
Max(A) is a compact Hausdorff topological space.

1.3 Tychonoff Spaces and Semi-Prime f-Rings

We will now look at the ring of continuous real valued functions. Let X be a

topological space. Denote by C(X) the set of all continuous real valued functions on
X. For f,g E C(X), define

1. f + g by (f + g)(x) = f(x) + g(x) for all x E X.

2. f g by (f g)(x) = f(x)g(x) for all x E X.

We will write fg for f g. These operations make (C(X), +, ) a commutative ring.
If in addition we define scalar multiplication by,

3. For r E R, rf by (rf)(x) = r(f(x)) for all x E X.

Then C(X) is a real vector space. We also define <, A and V by,

4. f < g if f(x) 5 g(x) for all x E X.

5. f A g by (f A g)(x) = min{f(x),g(x)}, for each x E X.


6. f V g by (f V g)(x) = max{f(x),g(x)}, for each x E X.










With these, C(X) becomes a lattice ordered ring and a vector lattice. In addition,

C(X) is an f-ring and C(X) has the bounded inversion property.
To distinguish topological spaces by the algebraic properties of C(X), the next

theorem will allow us to restrict our attention to the class of Tychonoff spaces. We

first need the following definitions.

Definition 1.3.1 Two subsets A and B of X are said to be completely separated if

there is an f E C(X) such that f(A) = 0 and f(B) = 1.

Definition 1.3.2 A Hausdorff space X is said to be a Tychonoff space if for any closed

set K and any x K, {x} and K are completely separated.

Theorem 1.3.1 (Theorem 3.9 [15]) For every topological space X, there exists a Ty-

chonoff space Y and a continuous mapping r from X onto Y such that the mapping
g + g o r is an isomorphism from C(Y) onto C(X)

This induced mapping is both a ring and a lattice isomorphism. For the remainder

of this paper, unless explicitly stated otherwise, "space" will mean "Tychonoff space".

For Y C X we will denote the closure of Y in X by clx(Y) or, when no ambiguity

will result, by cl(Y). Similarly we will denote the interior of Y in X by intx(Y) or

int(Y).

Definition 1.3.3 For f E C(X) let Z(f) = x E X : f(x) = 0}. This is called the
zero set of f. Let coz(f) = {x E X : f(x) $ 0}. This is called the cozero set of f.
Let Z(X) = {Z(f) : f E C(X)}.

Tychonoff spaces have the following useful characterization.

Theorem 1.3.2 (Theorem 3.2 [15]) A space X is a Tychonoff space if and only if

Z(X) is a base for the closed sets of X.










It will occasionally be helpful to use this result as: X is Tychonoff if and only if
the cozero sets are a base for the open sets of X.
What follows is a rudimentary description of the construction and properties of
the Stone-Cech compactification of a Tychonoff space. A more detailed account can
be found in Chapter 6 [15], or Chapter 1 [33].
Let X be a Tychonoff space and let Z(X) denote the zero-sets of X. F C Z(X)
is called a z-filter if,

1. 0 F.

2. IfZ Z, Z2 E F then Z1 n Z2 E F.

3. If Z F and Z C K C X then K E F.

A maximal z-filter is called a z-ultrafilter. Let fX denote the set of all z-ultrafilters
of X. For Z E Z(X) let Z = {a E fX : Z E a}. We topologize fX by taking

{Z: z E Z(X)} as a base for the closed sets. For a E fX, if na # 0 then there is an
x E X such that noa = {x}. The map x :- {a E fX :x E a} is a dense embedding
of X into 3X. It is a theorem due to Gelfand and Kolmogoroff [13] that the map
p '-* {f E C(X) : p E clpxZ(f)} is a one to one correspondence between the points

of pX and the maximal ideals of C(X). In fact, this map is a homeomorphism from

fX onto Max(C(X)). We will use this homeomorphism extensively.
Denote by C*(X) the set of all bounded continuous real valued functions on X.
Then C*(X) is an f-subring of C(X). We need to make the following definition.

Definition 1.3.4 Let X C Y be topological spaces. X is said to be C*-embedded in Y

if every f E C*(X) can be extended to a function in C(Y).

The following is most of Theorem 1.46 [33] and is a compilation of results char-

acterizing #X.










Theorem 1.3.3 Every Tychonoff space X has a unique compactification O3X which has

the following equivalent properties:

1. X is C*-embedded in f3X.

2. Every continuous mapping of X into a compact space Y extends uniquely to a

continuous map from fX into Y.

3. If Za and Z2 are zero sets in X, then clpxZ1 l clxZ2 = clx(Z1 n Z2).

4. Disjoint zero sets in X have disjoint closures in 3X.

5. Completely separated sets in X have disjoint closures in fiX.

6. 3X is the maximal compactification of X in the sense that if Y is any com-

pactification of X, then there is a continuous map from 3X onto Y that is the

identity on X.

Moreover, fX is unique in the sense that if T is a compactification of X .stisfying

any of the above conditions, then there is a homeomorphism of 3X onto T that is the

identity on X.


For the purposes of this dissertation, the major significance of Theorem 1.3.3 is

that in many cases, we can without loss of generality assume the space we are dealing

with is compact.

1.4 Boolean Duality

A lattice is a partially ordered set (B, <) such that for every pair {a, b} C B,

the least upper bound of a and b exists in B as does the greatest lower bound of a

and b. We will denote the least upper bound of a and b by a V b and call this the

join of a and b, and we will denote the greatest lower bound of a and b by a A b and










call this the meet of a and b. A lattice B is said to be bounded if it has a least and

largest element. In this section we will denote these, when they exist, by 0 and 1
respectively.

Definition 1.4.1 A bounded lattice B is said to be complemented if for every a E B

there is a b E B such that a A b = 0 and a V b = 1. Such a b is called a complement

of a.

Definition 1.4.2 A lattice B is said to be distributive if for every a, b, c E B,

a A (bV c) = (aAb) V (a A c).

Definition 1.4.3 A bounded distributive complemented lattice B is said to be a boolean

algebra.

In a boolean algebra, the complement of any element is unique and for a E B we

will denote the complement of a by a'. For a, b E B, we will denote the difference by

a \ b = a A b'. Boolean algebras have the following properties. A more complete list

can be found in Chapter 1 of "Boolean Algebras" [28] or Chapter 3 in [27].

Let B be a boolean algebra and let a, b, c E B. Then,

1. (a V b)' = a' A b' and dually.

2. 0' = 1, 1' = 0, (a')' = a.

3. a < b if and only if b' < a'.

4. a A b = 0 if and only if a b'; a V b if and only if a' < b.

5. For any finite {a : 1 < i < k} C B, a \ (V ai) = A(a \ a).

Definition 1.4.4 Let A, B be boolean algebras. A map 0 : A -- B is called a boolean

homomorphism if 0 is a lattice homomorphism and for every a E A, O(a') = (0(a))'.










We are now going to look at the construction of the Stone space of a boolean

algebra. This construction was originally carried out in M. Stone's 1936 paper, "Ap-

plication of the theory of boolean rings to general topology" [29]. Let B be a boolean

algebra and let 0 # F C B. F is called a filter on B or a B-filter if the following

hold;

1. 0 c F.

2. If a, b E then a A b E .

3. IfaE Fanda
A B-filter F is called a B-ultrafilter if it is a maximal B-filter; that is, if g is a

B-filter and F C G then F = G. B-ultrafilters have the following characterization.

See Section 2.3 [27].

Theorem 1.4.1 Let F be a B-filter. The following are equivalent.

1. 7 is a B-ultrafilter.

2. For every a E B, if a A b # 0 for every b F then a E F.

3. For every a E B, either a E T or a' E F.

For a boolean algebra B, let St(B) denote the set of B-ultrafilters. For b E B

let 14 = {F E St(B) : b F}}. We topologize St(B) by taking {Ub : b E B} as a

base for the open sets. Since we will see this kind of construction again, we will look

closely here at the relationship between intersection, union and complementation of

basic open sets in St(B), and meet, join and complementation of elements of B.

Let a, b E B. The following can be easily verified.


1. UaUb = baVb.











2. U U Ub = UaAb.

3. St(B) \ U, = U,.

4. U1 = 0 and Uo = St(B).

If F E St(B), since F is a B-ultrafilter, for 0 = b E B either b E F or b' E F.

Therefore either F Ub, or F E Ub. If F E Ua n b then a V b F so that F E Uavb*

By (1) above, {Ub : b E B} is a base for the open sets of a topology on St(B).
Let X be a Hausdorff space. A subset K of X is called clopen if K is both closed

and open. Let B(X) denote the set of clopen sets of X. With the operations of set

theoretic union (U), intersection (n) and complementation ('), (B(X), U, n,'), is a

boolean algebra.

Definition 1.4.5 A Hausdorff space X is said to be zero-dimensional ifB(X) is a base

for the open sets of X.

Definition 1.4.6 A Hausdorff space X is said to be totally disconnected if for every

x E X, the connected component of x is {x}.

We will make use of the following result which is Theorem 16.17 in [15].

Theorem 1.4.2 Let X be a Tychonoff space. Each of the following conditions implies

the next and if X is compact, all conditions are equivalent.

1. Disjoint zero sets are contained in disjoint clopen sets.

2. X is zero-dimensional.

3. X is totally disconnected.

We are now ready to state Stone's Representation Theorem. The following version

is Theorem 3.2 (d) [27].










Theorem 1.4.3 Let B be a boolean algebra. Then,

1. St(B) is a compact zero-dimensional Hausdorff space.

2. {Ub : b E B} = B(St(B)).

3. The map b F- b Ub is a boolean isomorphism from B onto 3(St(B)).

Categorically Let CZD denote the category having as objects compact zero-

dimensional Hausdorff spaces and continuous maps as morphisms. Let BA be the

category of boolean algebras and boolean maps.

Define F : CZD -+ BA by F(X) = B(X) for objects and if f : X -+ Y is a

morphism, define F(f) : B(Y) -- B(X) by: for K E B(Y), F(f)(K) = f-l(K). It

can then be shown that F is a contravariant functor from CZD to BA.

Define G : BA -+ CZD by G(B) = St(B) for objects and if 0 : A -- B is a

morphism, define G(O) : St(B) -- St(A) by: for F E St(B), G(O)(.) = 0-1'(). G
is a contravariant functor from BA to CZD. Moreover, as (3) from Stone's Theorem

may indicate, CZD and BA are dual categories.















CHAPTER 2
LOCAL-GLOBAL f-RINGS

2.1 Introduction

Let A be a commutative semi-prime f-ring with identity and bounded inversion.

We first need to recall some ideas from commutative ring theory (see Chapters 2 and

4 in "Multiplicative Ideal Theory" [16]).

A subset S of A is called a regular multiplicative closed subset if 0 0 S, S is

closed under multiplication and S contains no divisors of zero. Denote by S-1A

the set of all fractions a/s where a E A and s E S. If we define equality, addition

and multiplication as in the classical construction of the quotient field of an integral

domain, then S-1A is a commutative semi-prime ring with identity and the map

a F-+ (as)/s is a monomorphism from A into S-A. If 1 E S, we can take s = 1 and

this map becomes a i- a/I. S-1A is called the ring of quotients of A with respect

to S. If we take S to be the set of all regular elements of A, then S-1A is called the

classical ring of quotients of A and will be denoted by qA. We can define an ordering

on qA by first observing that for a/s we can assume that s > 0, since a/s = as/s2.

Then define (a/s) V 0 = (a V 0)/s. This extends the ordering on A in such a way that

qA is an f-ring and the embedding of A into qA preserves order. If a/s > 1 in qA,

then a > s and since s is regular so is a. Therefore, qA has bounded inversion.

We now look at the localization of A at a prime ideal P. Let O(P) = {a E

A : ab = 0 for some b 0 P}. Then it can be shown that O(P) is the intersection

of the minimal prime L-ideals contained in P and therefore that O(P) is an ideal

and an -ideal. We define Ap to be the ring of quotients of A/O(P) with respect

19










to S = {b + O(P) : b V P}. The point of factoring out O(P) is to insure that S

contains no divisors of zero. In the case where M is a maximal ideal, since A is an
f-ring with bounded inversion, A/O(M) is already a local ring having M/O(M) as

its unique maximal ideal. Therefore the elements of S are already invertible and so

AM A/O(M). In the context of commutative semi-prime f-rings with bounded

inversion, when we refer to the localization at a maximal ideal we will mean the

quotient ring A/O(M).

We need the following definitions.

Definition 2.1.1 A polynomial f E A[zx, ..., xn] is said to represent a unit over A if

there exists al,..., a, E A such that f(al,..., an) is a multiplicative unit in A.

For f E A[x1, ..., Xn], let fM denote the polynomial in AM[Xl, ..., z,] whose coeffi-

cients are the respective images of the coefficients of f under the map a i- a/1.

Definition 2.1.2 A ring A is said to have the local-global property if for each polyno-
mial f E A[xl,..., ,], whenever fM represents a unit over AM, for all M E Max(A),

then f represents a unit over A.

In their paper "Module Equivalences" [11], Estes and Guralnick look at many

of the implications of the local-global property. The authors first point out that

historically, many of the results obtained for local-global rings are extensions of results

first obtained by R. S. Pierce for rings which are Von Neumann regular modulo their

Jacobson radical [26]. In particular, for modules over local-global rings, M1 N _

M2 E N implies M1 M2, and Mn N' implies M N.
Recall that a polynomial f(x) E A[x] having coefficients ax,..., an is said to be

primitive if there exists bl,..., bn E A such that albl + + anb = 1.











Definition 2.1.3 A ring A is said to satisfy the primitive criterion if for any primitive

polynomial f E A[x], f represents a unit over A.


By a residue field we mean the field of quotients of the integral domain A/P for

P a prime ideal.

The following result, due to Estes and Guralnick [11], will allow us to determine

when an f-ring is local-global in terms of the primitive criterion.

Lemma 2.1.1 A ring A satisfies the primitive criterion if and only if A is a local-global

ring with all residue fields infinite.

Consequently, if A is a commutative f-ring then A is local-global if and only if

A satisfies the primitive criterion. We will use this characterization of local-global

f-rings to eventually characterize the local-global property in terms of Max(A).

Recall that for an f-ring A, B C A is said to be an f-subring if B is a subring and

a sublattice of A. We have the following theorem relating an f-subring of an f-ring

and their respective maximal spectra.


Theorem 2.1.1 Let A be a commutative semi-prime f-ring with identity and bounded

inversion. If B is an f-subring of A with bounded inversion and 1 E B then the map

0: Max(A) -- Max(B)

defined by O(M) = the unique maximal ideal containing M n B, is a continuous sur-

jection.


PROOF

We first note that for M E Max(A), M n B is a prime ideal and a prime t-ideal

of B. Therefore M n B C M' E Max(B) for some M' and as this is also a prime










e-ideal and the prime -ideals form a root system, M' is unique. Therefore, 0 is well
defined.
To show that 0 is onto, let N E Max(B). Since 1 N, there is a value V of 1 in B

with N C V. Let W be the -ideal generated by V in A; that is W = {a E A : lal < v
for some v E V}. Then 1 W so that W C W', a value of 1 in A. Let P be a
minimal prime -ideal contained in W'. Then P is a minimal prime ideal, so that

P C M for some M E Max(A). Since M is a prime f-ideal, P C M C W'. Now,
W' B is a prime e-ideal of B containing V with 1 W' n B. As V is a value

of 1, V = W' nB and V D M B. Since MnfB is a prime ideal and a prime
#-ideal of B it is contained in a unique maximal ideal O(M) of B with O(M) C V.
If N 0 O(M), then B = N + )(M) C N V O(M) C V. This is a contradiction.
Therefore, O(M) = N, so that 4 is onto.
To show that ) is continuous, we will consider 4 as the composite po o p where
L : Spec(B) -- Max(B) is defined by p(P) = the unique maximal ideal containing P
and p : Max(A) -+ Spec(B) is defined by p(M) = Mn B. To see that p is continuous,
let x E B and let A/a = {P E Spec(B) : x P}. This is a basic open set for the hull-
kernel topology on Spec(B). Since x E B C A, M = {M E Max(A) : x A} is a
basic open set for the hull-kernel topology on Max(A). It is easy to see that M E Mx

if and only if M n B E Af, so that p-1(A',) = M, and p is continuous. To show that
p is continuous, we will show continuity at an arbitrary point. This part of the proof
is modeled on Lemma 10.2.3 [5]. Let P E Spec(B) and suppose that U is an open

neighborhood of p(P) in Max(B). Without loss of generality, we may assume that U
is a basic open set in Max(B). Say U = M,= {M E Max(B) : x M}. Then U =

Max(B)nAN, where NA = {P E Spec(B) : x ( P} is a basic open set in Spec(B). Let

Q E Max(B) \A' Then Q and p(P) are distinct maximal ideals. Since Q and j(P)
are also t-ideals, we can find 0 < x E Q \ L(P) and 0 < y E p(P) \ Q with x A y = 0,










so that Q and t(P) are contained in disjoint open sets in Spec(B). Suppose Q E UQ
and p(P) E VQ, both open with Ug n VQ = 0. Consider {UQ : Q E Max(B) \ NAf},
where the UQ are as above. Since any neighborhood of a maximal ideal M in Spec(B)

is a neighborhood of any prime P C M, {UQ : Q E Max(B) \ AJ'} is an open cover
of Spec(B) \ Now, Spec(B) \ is a closed subspace of of Spec(B) and so is
compact. Therefore there exists {Qi, 1 < i < n}, such that Spec(B) \ANx C U='i UQi.
Let V = nf=1 VQ,, where the VQ, are as above. Then p(P) E V so that P E V and
V C .A. Then p(V) C (.NA) = .MA and therefore, p is continuous.
We have shown that = p o p and that this is a continuous surjection from
Max(A) to Max(B).
QED

Of particular interest in what follows will be the occasions when the map 0 defined
in the proceeding theorem is one-to-one. To describe these occurences we make the
following definition.

Definition 2.1.4 Let A and B be as in the statement of Theorem 2.1.1, and let q be
the continuous surjection guaranteed by the theorem. If 0 is a homeomorphism, we
say that B separates the points of A.

2.2 The Specker Subring of a Semi-Prime f-Ring

For A an f-ring, let A(1) denote the set of bounded elements of A. That is
A(1) = {a E A : Ja| < n 1 for some n E N}. Then A(1) is an f-subring and an
-ideal of A. If A = A(1) we say that A is bounded. We have the following corollary
to Theorem 2.1.1.

Corollary 2.2.1 Let A be a commutative semi-prime f-ring with identity and bounded
inversion. Then Max(A) 2 Max(A(1)).










PROOF

We first note that if A has bounded inversion then so does A(1), so that both
Max(A) and Max(A(1)) are compact Hausdorff. Since A(1) is an f-subring of A

with 1 E A(1), by Theorem 2.1.1, it suffices to show that the map 0 : Max(A) -+

Max(A(1)) is one-to-one. Recall that for M E Max(A), O(M) is the unique element

of Max(A(1)) containing M n A(1). Suppose that N, M E Max(A) with N i M. If

O(N) = O(M), since A(1) E C(A) and C(A) is distributive, we have that

O(M) A()) (NA()) A()) = (NVM)nA(1) = (N+M)nA(1) = AnA(1) = A(1)

This is a contradiction. Therefore, O(N) j O(M) so that is one-to-one.
QED


For A a semi-prime f-ring with bounded inversion, if n E N, then n 1 > 1 in A

so that (n 1)-1 E A. In particular, A is divisible, so that we may consider A as an

algebra over Q. Let S(A) denote the Q subalgebra generated by the idempotents of

A. We will call this the speaker subring of A. If A is a vector lattice, the specker

subring of A is S3(A) the real subalgebra generated by the idempotents of A.


Definition 2.2.1 An -group G is called hyper-archimedean if every i-homomorphic

image is archimedean.

We will call an f-ring A hyper-archimedean if its -group structure is hyper-

archimedean. Most of following theorem was obtained originally for vector lattices,

and is due in this more general form to P. Conrad [8].










Theorem 2.2.1 Let G be an (-group. The following are equivalent.

1. G is hyper-archimedean.

2. Each proper prime i-ideal is maximal and hence minimal.

3. G = G(g) E g9 for each g E G.

4. G is f-isomorphic to an -subgroup G* of nJ Ri and for each 0 < x, y E G* there
exists an n > 0 such that nxi > yi for all xi = 0.

5. IfO < a, bEG then aA nb = aA (n + 1)b for some n > 0.

We also have the following characterization which will be useful in the present

context.

Theorem 2.2.2 (Theorem 14.1.7 [5]) An -group G is hyper-archimedean if and only

if G is -isomorphic to an -subgroup of C(X) such that G separates the points of X

and the support of each g E G is compact and open.

Since S(A) is archimedean and has a strong order unit, by Yosida embedding,

S(A) embeds as an -subgroup of C(Yos(S(A), 1)) and S(A) separates the points

of Yos(S(A)). Since each a E S(A) is a finite sum of idempotents and idempotents

must map to the characteristic functions of clopen sets, the support of a is clopen

hence compact and open. We then have most of the following lemma.

Lemma 2.2.1 For A a commutative semi-prime f-ring with identity and bounded in-

version, S(A) is hyper-archimedean. If A is also a vector lattice then SR(A) is hyper-

archimedean.

Theorem 2.2.3 Let A be a commutative semi-prime f-ring with identity and bounded

inversion. If S(A) is convex, then A is hyper-archimedean and A = S(A).










PROOF

Suppose that S(A) is convex in A. By Theorem 1.1.6, the map P -* P f S(A)

is a one to one order preserving correspondence between the prime -ideals of A

not containing S(A) and the proper prime -ideals of S(A). Since S(A) is hyper-

archimedean, the prime -ideals of S(A) are trivially ordered and consequently, so

are the prime e-ideals of A which do not contain S(A). We will show that no prime

-ideal of A contains S(A).

Let P be a prime e-ideal of A. Then P C V a value of 1. V D Q a minimal

prime f-ideal and since A is semi-prime, Q is also a prime ideal and so is contained

in a maximal ideal M. Since A has bounded inversion, M is an prime t-ideal and so,

Q c M C V and Q c P C V. We will show that P C M. Now consider the quotient

ring A/M. Since M is both a maximal ideal a prime e-ideal, A/M is an ordered field.

Since S(A) and M are convex, (S(A)+ M)/M is a convex f-subring of A/M. Also, as

1 ~ M, M h S(A) so that M n S(A) is a proper prime f-ideal of S(A). Since S(A) is

hyper-archimedean, S(A)/(S(A)nM) is archimedean. By Theorem 2.3.9 [5], we have

that S(A)/(S(A) n M) (S(A) + M)/M as i-groups and hence as f-rings. Suppose

now that x > 1 in A/M. Then x-1 < 1, but 1 c (S(A)+M)/M and this is convex and

archimedean; a contradiction. Therefore, A/M is a real field, in particular VIM = 0

so that V = M. Then P C M and since M 6 S(A), we have that P ; S(A). We

have shown that any prime i-ideal does not contain S(A) and therefore the prime t-

ideals of A are trivially ordered. By Theorem 2.2.1, A is hyper-archimedean. Clearly,

S(A) C A(1). Since 1 E S(A) and S(A) is convex, A(1) C S(A). Since A is hyper-

archimedean with identity, by Lemma A [8], A = A(1). Therefore A = S(A).
QED










We should note that if A is, in addition, a vector lattice, by the above and

Proposition 1.2 [8], the following are equivalent.

1. A is hyper-archimedean.

2. A = S(A).

3. Sa(A) is convex in A.

We will also need the following lemma.


Lemma 2.2.2 If A is a commutative semi-prime f-ring with identity and bounded

inversion, then so is S(A). If A is a vector lattice this holds for S,(A).


PROOF

That S(A) is a commutative semi-prime f-ring with identity follows from S(A)

being an f-subring of A and 12 = 1. We need only show that S(A) has bounded

inversion. Let a E S(A). Then a = E?=1 qjei where for 1 < i < n, qi E Q and e, E A

is idempotent. We first note that a can be written as a = 71i rifi where the ri E Q

are distinct and more importantly, the fi E A are idempotents such that fifj = 0 for

i 0 j. Suppose now that 1 < a E S(A). Then there is a b E A such that ba = 1. That

is b(ET=I rifi) = ETi brifi = 1. For each 1 < i < m we get that br;f, = fi, so that

Ei1 fi = 1. Let c = 1(())f/. Then c E S(A) and ac = E 1 fi = 1. Therefore,
S(A) has bounded inversion. The proof is similar for Si(A), taking real coefficients

instead of rational.
QED










2.3 A Characterization of Local-Global f-Rings

The main result of this section is the following theorem.


Theorem 2.3.1 Let A be a commutative semi-prime f-ring with identity and bounded

inversion. Each of the following conditions implies the next one.

1. A is local-global.

2. Ifa bt2 E A[t] is primitive with positive coefficients, then a bt2 represents a

multiplicative unit in A.

3. Max(A) is zero-dimensional.

4. Max(A) ^ Max(S(A)). (If A is a vector lattice, Max(A) Max(Sa(A)).)

Furthermore, (2), (3) and (4) are always equivalent and if A is bounded, then all four

conditions are equivalent.

We proceed with a series of lemmas which will be used in the proof of this theorem.

Lemma 2.3.1 (Proposition 2.1 [9]) Min(A) is a zero dimensional Hausdorff space.


For a ring A, let J(A) denote the Jacobson radical of A. Then J(A) = ({M:

M E Max(A)}. We have the following lemma.

Lemma 2.3.2 Let A be a commutative semi-prime f-ring with identity and bounded

inversion. Then Max(A) Max(A/J(A)).











PROOF

We first observe that since Max(A) C C(A), J(A) is an G-ideal of A so that

the canonical map A -+ A/J(A) is an -homomorphism as well as a ring homo-

morphism. Consequently, A/J(A) is a commutative semi-prime f-ring with iden-

tity and bounded inversion. The canonical map A i-+ A/J(A) induces a one-to-

one correspondence between the maximal ideals of A and the maximal ideals of

A/J(A). Let 0 : Max(A) Max(A/J(A)) be this induced map. Since Maz(A)

and Max(A/J(A)) are both compact Hausdorff spaces, we need only show that 0 is

continuous. Let M = {M E Max(A) : x M} be a basic open set of Max(A).

Since M D J(A) for every M E Max(A), we have that

M E M, x M = 7 9 0(M) O(M) E M-,

where x E A and 7 E Max(A/J) are such that x i-- 7 under the canonical map,

and My = {M E Max(A/J) : T M M}. Thus 0-1(MA) = M.. Therefore, 0 is

continuous.
QED


We have seen that the maximal spectra of A and A/J(A) are homeomorphic. We

now consider some algebraic properties of A and A/J(A).


Lemma 2.3.3 Let A be a commutative semi-prime f-ring with identity and bounded

inversion. Then A is local-global if and only if A/J(A) is local-global.


PROOF

Let p(t) = ao + alt + + ant" E A[t] and let p(t) = 0 + -at + ** + Ent" E

(A/J(A))[t] where ai and ai are such that ai H-+ di under the canonical map A -- A/J.
By Lemma 2.1.1, we need only show that p(t) is primitive if and only if p(t) is. It

is clear that if p(t) is primitive, then so is p(t). Suppose now that p(t) is primitive.











Then there exists bo,1, -, b A/J(A) such that aobo + albl + + ab = 1 + j

for some j E J(A). If aobo + alb + + a,b, = 1 + j E M for some M E Max(A),

then 1 = aobo + albl + + a,b, j E M as j E J(A) C M. This is a contradiction.

Therefore, aoboa + + +a,bn is a unit, so that p(t) is primitive and the result

follows.
QED



Lemma 2.3. Let A be a commutative semi-prime f-ring with identity and

bounded inversion. Then Yos(A, 1) 2 Max(A). If in addition A is bounded then

Yos(A, 1) = Max(A).


PROOF

For each M E Max(A), M is a prime e-ideal with 1 ( M and so it is contained

in a unique value V E Yos(A, 1). For V E Yos(A, 1), V D P a minimal prime f-ideal

and a minimal prime ideal. Then there is an M E Max(A) with P C M C V. If

Mi = M2 Max(A) with M, M2 C V then A = Mi + M2 C M1 V M2 C V, so that

each value of 1 in A contains exactly one maximal ideal. As in the proof of Theorem

2.1.1, this correspondence induces the required homeomorphism.

Suppose that A is bounded. Since the topologies on both spaces are their respec-

tive hull-kernel topologies, it suffices to show that Yos(A, 1) = Max(A) as sets. Let

Y e Yos(A, 1). Y D P a minimal prime -ideal, hence a minimal prime ideal. Then

P C M E Max(A) with P C M C Y. Now consider the quotient ring AIM. Since

M is a maximal ideal, AIM is a field. Since M is also a prime -ideal, AIM is an or-

dered field. Since A is bounded, AIM is as well. Therefore, AIM is order-isomorphic

to a subfield of R. Now, Y/M is an e-ideal of A/M and so by Holder's Theorem,










Y/M = 0. Therefore, Y = M, so that Y is a maximal ideal of A. We then have that

Yos(A, 1) C Max(A). By the above homeomorphism, Yos(A, 1) = Max(A).
QED


For the next result we will need the following definition.


Definition 2.3.1 Let G be an -group. For 0 < a, b E G we say that a is infinitesimal

to b, denoted a < b, if na < b for all n E N.


Lemma 2.3.5 Let A be a bounded commutative semi-prime f-ring with identity and

bounded inversion. Then J(A) = {x E A : |lz < 1} and A is archimedean if and

only if J(A) = 0.

PROOF

Let x E J(A). Since J(A) is convex, we may assume that x > 0. Suppose now

that for some n E N, nx 1. Then (nx 1)+ > 0, so there is a minimal prime -ideal

P with (nx 1)+ > 0 mod P. Since P is prime, (nx 1)- = 0 mod P, so that

(nx 1) > 0 mod P and nx > 1 mod P. Now, P is a minimal prime ideal so there
is an M E Max(A) with P C M. The canonical map a + P a + M is a lattice

homomorphism from A/P to A/M so that nx > 1 mod M. But, x E J(A) C M and

M is convex so that 1 E M; a contradiction. Therefore, x < 1.

Suppose now that lzx < 1. Since each M E Max(A) is a prime i-ideal, nixI <

1 mod M for all M E Max(A), n E N. By Lemma 2.3.4, each M E Max(A) is a value

of 1 in A. Since A is bounded, A/M = M*/M is archimedean so that lx| = 0 mod M,

whence x E M. Since this holds for each M E Max(A), we have that x E J(A).

Clearly, if A is archimedean, then J(A) = 0. Suppose now that J(A) = 0. Let

0 < a, b E A and suppose that na < b for all n E N. Then na < b < bV1 = c

for all n E N. Since c > 1 and A has bounded inversion, c-1 E A. Then (na)c-1 =










n(ac-') < 1 for all n E N. Therefore, ac-1 E J(A) = 0, so that a = 0, and A is

archimedean.
QED


It should be pointed out for later use that the containment J(A) C {x E A :

lxi << 1} obtains without the assumption that A is bounded.
We will need the following lemma which characterizes the clopen subsets of

Max(A).


Lemma 2.3.6 Let A be a commutative semi-prime f-ring with identity and bounded

inversion. Then KC C Max(A) is clopen if and only if KC = MA where x E A is

idempotent.


PROOF

Suppose first that KS = M. where x E A is idempotent. Then Mx-1 is a basic

open set disjoint to M. with Mx-1 U M. = Max(A). Therefore Mx is clopen.

Suppose now that KS C Max(A) is clopen. Since KS is open, K U = UJxB M, for some

B C A. Since K) is a closed subset of Max(A), it is compact and therefore there

exists {x( : 1 < i < n} such that K = UF= M,,. Since each M E Max(A) is convex,

we may assume that each xi > 0. Then KS = U=l M,= = M, where x = V=i xi.

By similar argument, since KS is clopen, Max(A) \ K = My for some 0 < y E A.

Now M, n My = MA = 0 so that xA y E M for every M E Max(A). Let

' = x (xA y) and y'= y (xA y). Then M = MT,, My = My and A y'= 0.

Since Mx, U My, = Mxvy, = Max(A), x' V y' is a multiplicative unit. Now let

e = x'(x' V y')-' and f = y'(x' V y')-1. Then e V f = 1, e A f = 0 and Me = Mx.

Since eAf = 0, ef = 0 so that e2 = e2V = e2Vef = e(e V f) = e() = e. Therefore,

K = Me where e is idempotent.
QED











We now proceed to the proof of the main theorem.

PROOF

The proof will consist of two parts. We first will show that (1) implies (2); that

(2) implies (3); that (3) is equivalent to (4), and that (3) implies (2). We will then

show that if A is bounded that (3) implies (1).

Clearly, (1) implies (2). Suppose now that (2) holds. Since Max(A) is compact,
by Theorem 1.4.2, it suffices to show that Max(A) is totally disconnected. Let

N, M E Max(A) with N 5 M. Since A has bounded inversion, both M and N are
-ideals. Pick 0 < x E M \ N. By the maximality of N, the ideal generated by x and
N, (N, x) = A. In particular, there exists 0 5 y E N, a E A such that y + ax = 1.

Then y M, and (y V 0) + ax = (y + ax) V (0 + ax) 2 y + ax = 1 so that by bounded

inversion, (y V 0) + ax = u is a multiplicative unit in A. By the convexity of N and

M, y V 0 E N \ M. By a suitable relabelling, we have produced 0 < x E M \ N,
0 < y E N \ M with (x,y) = 1. Now consider x yt2 E A[t]. This satisfies the
hypothesis of (2) and so there is an a E A so that x-ya2 = u is a multiplicative unit in

A. Let u+ = uV0 and u- = (-u)V0. Then u = u+ u- and u+ Au- = 0. Recall that

the basic open sets of Max(A) are of the form M.x = {M E Max(A) : x M} for

x E A. We now note that since the maximal ideals are t-ideals, if 0 < y E A, then

Mx n My = MaAy and M, UMy = MV2y. Now, u+ Vu- > u and as A has bounded

inversion u+ V u- is a multiplicative unit. Then .M+ U Mu- = Mu+v- = Max(A)

and M/+ n M.- = M,+Au- = 0. Therefore {M,+,.Mu-} is a clopen partition of
Max(A). Now, -ya2 < x ya2 < x so that 0 < u+ < x and 0 < u- < ya2. Then

as x E M and M is convex, u+ E M so that u- 4 M and therefore M E MA-.

Similarly, since y E N, it follows that N E M,+. Thus, for any two distinct points
in Max(A) there is a partition of Max(A) into two clopen subsets each containing

exactly one of the points. Therefore, Max(A) is totally disconnected.










Suppose now that (3) holds. That is, Max(A) is zero-dimensional. Let :
Max(A) -- Max(S(A)) be the map defined in Theorem 2.1.1. Then is continuous
and onto, and since S(A) is hyperarchimedean, O(M) = M M S(A) for all M E
Max(A). Since both of Max(A) and Max(S(A)) are compact Hausdorff, it suffices

to show that 4 is one-to-one. Let M $ N E Max(A). Since Max(A) is zero-
dimensional, hence totally disconnected, there is a clopen set /C C Max(A) with

M E KC, N )C. Since CK is clopen, by Lemma 2.3.6, there is an e E A, idempotent
such that CK = Me. Since e is idempotent, e E S(A) so that e n M n S(A) and

e E N n S(A). Therefore O(M) 7 O(N), and therefore is one-to-one.
If Max(A) Max(S(A)) then, since S(A) is hyper-archimedean, Max(S(A)) =
Min(S(A)), and Min(S(A)) is zero-dimensional, so that Max(A) is as well.
That (3) implies (2) is a special case of the second part of the proof. Suppose it has

already been shown that if A = A(1) then all four conditions are equivalent. Suppose
now that (3) holds, that is Max(A) is zero-dimensional and A is not necessarily

bounded. Let a bt2 E A[t] be primitive with positive coefficients. Then there are

c, d E A such that ac + bd = 1. Then I1| = Jac + bdl < |ac| + IbdI = aic| + bldl <

(a+b)lcl+(a+b)ldl = (a+b)(lcl +ld). Since A has bounded inversion, (a+b)(Icl +dl)
is a unit, so that (a + b) is as well. Let a' = a(a + b)-1 and b' = b(a + b)-1. Then
a'- b't2 e A(1)[t] and a'+ b' = 1 so that a'-b't2 is primitive with positive coefficients

in A(1). If Max(A) is zero-dimensional, by Corollary 2.2.1, Max(A(1)) is also.

Therefore there is an x e A(1) such that a'- b'x = u is a multiplicative unit in A(1).
Since A(1) C A, x E A and u is a multiplicative unit in A. Therefore a-bx2 = u(a+b)

is a multiplicative unit in A.

Now let us prove that (4) implies (1) for bounded rings. By applications of
Lemmas 2.3.1, 2.3.2, 2.3.4, we may without loss of generality assume that A is

archimedean. By Lemma 2.3.4, Yos(A, 1) = Max(A) and by the preliminaries










on the Yosida embedding of an archimedean f-ring, A embedds as an f-subring
A C D(Max(A)), where A is an f-ring such that 1 I- 1. Since A is bounded,
the Yosida embedding is actually into C(Max(A)). Let Max(A) = X. Identifying
A with its image in C(X), we have that A is an f-subring of C(X) with 1 E A.
Let ao + alt + + at" E A[t] be primitive. If we consider the ai as elements of

C(X), we have that nfl=L Z(ai) = 0. Then {coz(ai) : 1 < i < n} is a basic cover
of X (in the sense of Chapter 16 [15]). Since X is compact and zero-dimensional,
there exists a refinement of this basic cover by a partition of X. That is, there ex-
ists {KIi : 1 < i < n}, a pairwise disjoint collection of clopen subsets of X, with
Ki C coz(ai) and such that U!I, K; = X.
Define f : Ki x R --+ R by f(x, r) = ao(x) + ai(x)r + + an(x)r'. Fix xo E KIi.
Then as Ki C coz(ai), ai(x) 0, so that x E ai-1(R \ {0}). Let fro : R --+ R be
defined by fxo(r) = f(xo,r). Then fo is continuous, so that f,0-j(R \ {0}) is a
non-empty open subset of R. Therefore there is an 0 / ro E f, -1(R \ {0}) n Q.
Now define fo : KCi R by fro() = f(x, ro). Then fro is continuous, and by
construction, fro(xo) 0 O. Therefore, coz(fr) # 0. Recall that ro is determined
by xo, so for ease of notation, let coz(fro) = N~o. Then {N x : x E K:i} is an
open cover of Ki. Since Ki is compact, there exists {N,. : 1 < j < m} such that
Ki = UTj NZ,. Since KC; is a clopen subset of a compact zero-dimensional space X,
Ki is a compact zero-dimensional space. Therefore there is a refinement of the basic
cover {N., : 1 < j < m} by a partition of Ki. Let {j : 1 < j < m} be this partition.
Recall that X = Max(A). Since each j is clopen in X by Lemma 2.3.6, there is in
ej E A idempotent such that Ej = Me,. In the identification of A with A, ej = XMe
Define si : X -+ R by si(x) = rlei(x) +. +rmem(x). Then si is continuous. Repeat
this construction for each Ki, and put s = sl + + s,. Then s E C(X) and s is a











linear combination of rational multiples of idempotents of A. Since A has bounded

inversion, s E A.

Now consider ao + als + + ans" as an element of C(X). By the construction

of s, Z(ao + als + .* + ans") = 0. If as an element of A, ao + als + + ans" E M

for any M E Max(A), then the Yosida embedding of A in C(X) would give us

an x E X, namely M, with ao(x) + al(x)s(x) + ..* + an(x)s"(x) = 0. Therefore

ao + als + + ans" M for every M E Max(A), so that ao + als + + a,sn is a

unit in A. Therefore A is local-global.

In the case that A is a vector lattice, nothing is changed by taking Sa(A) in the

place of S(A).
QED


We should note here that one interesting aspect of Theorem 2.3.1 is that condition

(2) is a first order condition in the language of commutative semi-prime rings with

identity; that is, condition (2) can be written in terms of universal and existential

quantifiers and a finite number of elements, operations and relations.

We should also note that in the above proof for A archimedean, bounded, the

substitution s E A such that ao + als + .* + ans" is a unit was an element of S(A).

This still holds if we drop the assumption of archimedeaneity.


Corollary 2.3.1 If A is a bounded commutative semi-prime f-ring with identity and

bounded inversion which is local-global and ao + alt + + ant' E A[t] is primitive,

then there is an s E S(A) such that ao + als + + ans" is a unit in A.


PROOF

It suffices to show that an idempotent in A/J(A) lifts to an idempotent in A.

That is, if x2 + J(A) = x + J(A), then there is an e E A idempotent such that

x + J(A) = e + J(A). So, suppose that x E A is such that x2 + J(A) = x + J(A).











Then x2 x = x(x 1) E J(A), so that x(x 1) E M for every M E Max(A). Since

M is a proper prime ideal this gives us that x E M or x 1 E M, but not both.

Therefore, Mx and Mx-1 partition Max(A), and M, is clopen. By Lemma 2.3.6,

M. = Me where e E A is idempotent. Suppose that x + J(A) = e + J(A). Then

there exists an M E A such that x e 0 M. Since M., = Me we have that x 0 M

and e ( M. But x(x 1),e(e 1) E M, so x 1,e 1 E M. A contradiction as

x e = (x 1) (e 1) M. Therefore x + J(A) = e + J(A).
QED



Corollary 2.3.2 Let A be a bounded commutative semiprime f-ring with identity and

bounded inversion and let B be an f-subring of A with bounded inversion and 1 E B.

If A is local global and S(A) C B, then B is local-global.


PROOF

Since A is bounded and B is an f-subring of A, B is bounded. By Theorem 2.3.1,

Max(A) Max(S(A)). Let 0 be this homeomorphism. By Theorem 2.1.1, and

Lemma 2.2.1, the maps 0 : Max(A) -- Max(B) and 0 : Max(B) -+ Max(S(A))

are continuous surjections with 0 o 0 = 0. Therefore i is one-to-one and hence a

homeomorphism. Then Max(B) is zero-dimensional and so, by Theorem 2.3.1, B is

local-global.
QED



Corollary 2.3.3 If A is a commutative semi-prime f-ring with identity and bounded in-

version which is in addition archimedean, bounded and local-global, then C(Max(A))

is the largest bounded archimedean local-global extension B of A such that A separates

the points of B.











PROOF

We first note that for A as above, C(Max(A)) is a commutative semi-prime f-ring

with identity and bounded inversion which is in addition archimedean, bounded and

local-global. Since Max(A) is compact Hausdorff, Max(C(Max(A))) 2- /(Max(A)) -

Max(A), so that A separates the points of C(Max(A)). Suppose now that B is

a bounded, archimedean, local-global extension of A with Max(A) Max(B).

Then as in the proof of Theorem 2.3.1, B is an f-subring of C(Max(B)). Since

Max(A) 2 Max(B), the result follows.
QED


Now let A = C(X) be the ring of continuous real valued functions on X. We will

denote S~(C(X)) by S(X). We make the following observations.

1. C(X) is a commutative semi-prime f-ring with identity and bounded inversion.

2. If X is compact, then C(X) is bounded.

3. If X is compact Hausdorff, then X O 3X 4 Max(C(X)).

These, together with Theorem 2.3.1, give us the following corollary.


Corollary 2.3.4 If X is a compact Hausdorff space, the following are equivalent:

1. C(X) is local-global.

2. If f-gt2 E C(X)[t] is primitive with positive coefficients, then f-gt2 represents

a unit in C(X).

3. X is zero-dimensional.


4. X Max(S(X)).











One of the troublesome aspects of the above results is that we get equivalence

of the stated conditions for bounded rings or in the case of C(X), for X compact.

We now consider a class of rings for which the assumption of boundedness can be

dropped. We need the following definitions.


Definition 2.3.2 A ring A with identity is said to be a Bezout ring if every finitely

generated ideal is principal.


Definition 2.3.3 A Tychonoff space X is said to be an F-space if every dense cozero

is C* embedded.


Bezout rings and Bezout domains are discussed extensively throughout R. Gilmer's

"Multiplicative Ideal Theory" [16]. The following is due to Gillman and Henriksen

[14] and in the present form is Theorem 14.25 [15].


Theorem 2.3.2 Let X be a Tychonoff space. Then the following are equivalent.

1. X is an F-space.

2. pX is an F-space.

3. C(X) is a Bezout ring.

4. 0 < a < b E C(X) implies that a = bf for some f E C(X).

5. Every ideal in C(X) is convex.

6. Every ideal in C(X) is an t-ideal.


7. For < a, b C(X), the ideal (a,b) = (a + b).










8. For f E C(X), there exists a k E C(X) such that f = kIf .

9. The localization of C(X) at any maximal ideal is a valuation ring.


We recall here that an integral domain D is called a valuation ring if the ideals

of D form a chain. Recall also that for A a commutative semi-prime f-ring with
identity and bounded inversion we have identified the localization of A at a maximal

ideal M with the quotient ring A/O(M). It is in this sense that (9) above should

be interpreted. The following is a partial generalization of the above to semi-prime

f-rings with bounded inversion [25].


Theorem 2.3.3 Let A be a semi-prime f-ring with bounded inversion. Then the fol-

lowing are equivalent.

1. A is a Bezout ring.

2. 0 < a < b E A implies that a = bf for some f E A.

3. Every ideal of A is convex.

4. Every ideal of A is an e-ideal.

5. For 0 < a, b A, the ideal (a,b) = (a + b).

6. The localization of A at any maximal ideal is a valuation ring.


We should note that if A is Bezout, then by (5) above, (f+, f-) = (f+ + f-) =

(If ). So that in particular, f = kifI for some f E A. For Bezout rings, Theorem
2.3.1 improves to the following.

Theorem 2.3.4 Let A be a commutative semi-prime f-ring with identity and bounded

inversion. If in addition, A is a Bezout ring then the following are equivalent.










1. A is a local-global ring.

2. If a bt E A[t] is primitive, then a bt represents a unit.

3. Max(A) is zero-dimensional.

PROOF

Clearly (1) implies (2). Suppose now that (2) holds. Since Max(A) is compact,
it suffices to show that Max(A) is totally disconnected. Suppose that M f N E

Max(A). As in the proof of Theorem 2.3.1, there exists 0 < a E M \ N and

0 < b N \ M with a A b = 0. Let f = a b. Then f+ = a and f- = b.

By Theorem 2.3.3, since A is Bezout, there is a k E A such that f = kifl. Then

a b = k(a + b) = ka + kb, so that (1 k)a = (1 + k)b. Since a A b = 0, we have

that (1 k)a = (1 + k)b = 0, and so a = ka and -b = kb. Let g = 1 k2. Then

ga = gb = 0, so that g E N and g E M. Also, g + k = 1, so by the hypothesis
there is a v E A such that k gv = u is a multiplicative unit. Consider now the

basic open sets Mu+ and Ms-. Since u is a multiplicative unit, these sets partition

Max(A). Now, u+a= (u V )a= ua V = (k- gv)a VO= kaVO= aVO=a,so

that u+ N lest a E N. Similarly, u-b = b, so that u- V M. Therefore N E M,+

and M E Ms-, and as these are disjoint, Max(A) is totally disconnected.

Suppose now that (3) holds. Let ao + a+t + + a4t" E A[t] be primitive. Then
there exists bo,bl,...,bn E A such that aobo + albl + -gg + a,b, = 1. Then

1 = laobo + alb + + anbl (laol + la +- + lan)(Ibo + bi + + Ibnl). Since

A has bounded inversion, aol + lal + + lan = a is a multiplicative unit. Then,
aoa-1 + ala-'t + + ana-lt' E A(1)[t]. Since A is Bezout, for each ai, 0 < i < n,

there is a ki such that ai = k|lai|. But then kiai = lail and we may assume without

loss of generality that Ikl < 1 (take k' = (kAl)V-1). Then koaoa-l+klala-+...*

knaa-1 = 1 and therefore aoa-1 a+a-at +.. + ana-lt" E A(1)[t] is primitive. Since










Max(A) is zero-dimensional, by Corollary 2.2.1, Max(A(1)) is as well. Applying

Theorem 2.3.1 to Max(A(1)) we have that A(1) is local-global. Therefore, there

exists a v E A(1) such that aoa-1 + ala-1v + + ana-lv' = u is a multiplicative

unit in A(1) C A. Therefore ao + alv + .. + a,v" = au is a multiplicative unit in A.

Since v E A(1) C A, A is local-global.
QED


For X a Tychonoff space we have the following corollary.

Theorem 2.3.5 If X is an F-space, then the following are equivalent

1. C(X) is local-global.

2. If f gt E C(X)[t] is primitive, then it represents a unit.

3. X is strongly zero-dimensional.

PROOF

By Theorem 2.3.2, X is an F-space if and only if C(X) is a Bezout ring. That (1)

implies (2) and that (2) implies (3) then follows directly from Theorem 2.3.4 and the

observation that Max(C(X)) OX. To prove that (3) implies (1), suppose that X

is strongly zero-dimensional; that is OX is zero-dimensional. Then C(PX) is local-

global. But, C(f3X) C*(X) the ring of bounded continuous real valued functions

on X. The proof now follows as in the proof of Theorem 2.3.4 taking C(X) = A and

C*(X) = A(1).
QED


Recall in the statement of Theorem 2.3.1, that conditions (2), (3) and (4) are

always equivalent and that (1) implies (2). The assumption of boundedness was

needed only in the proof that (3) implies (1). The problem that prevented this

implication in the general case is one of "cutting down" a primitive polynomial in











f(t) E A[t] to a polynomial f(t) E A(1)[t] and maintaining primitivity. The most
general result in this context is the following theorem.


Theorem 2.3.6 If A is a commutatuve semi-prime f-ring with identity and bounded

inversion, then the following are equivalent.

1. Every primitive polynomial f(t) E A[t] having coefficients that are comparable

to zero represents a multiplicative unit.

2. Max(A) is zero-dimensionl.


PROOF

It suffices to show that a primitive polynomial having coefficients that are compa-

rable to zero in A[t] differs from a primitive polynomial in A(1)[t] by a multiplicative

unit of A. We can then apply Theorem 2.3.1. The proof now proceeds exactly as in

the proof of Theorem 2.3.4 taking the ki to be 1 if a, > 0 or -1 if a, < 0.


QED















CHAPTER 3
QUASI-SPECKER f-RINGS AND f-RINGS OF SPECKER TYPE

3.1 Introduction

In the previous chapter, we have seen that if S(A) (respectively Si(A)) is a convex

f-subring of the commutative semi-prime f-ring A, then (i) S(A) = A (respectively

SS(A) = A), and (ii) A is local-global. In this chapter we will explore two other

distinct types of containments of S(A) in A and the effects of these containments on

the structure of A. One of these containments is based on the ring structure and the

other on the order structure of A. We need the following definitions.

Definition 3.1.1 Let R be a commutative ring. A subring S of R is said to be large

if for every non-zero S-submodule I of R, I n S 5 0. R is also called an S-essential
extension of S.

Definition 3.1.2 Let H be an t-group, G an e-subgroup of H. G is said to be large in

H if for every non-zero convex t-subgroup C of H, C n G $ 0. H is also called an

essential extension of G.

Since the above definitions become ambiguous in the context of f-rings, we will

adopt the convention that whenever using definitions dependent on the lattice struc-

ture we will precede them by an "o". Thus if A is an f-ring and B is an f-subring

such that B is large in A as an t-subgroup, we will say that B is o-large in A, or that

A is an o-essential extension of B.

We will make extensive use of the following well known characterization of essen-

tial extensions.











Theorem 3.1.1 Let R be a commutative ring, S a subring of R. Then, S is large in

R if and only if for every 0 $ a E R, there is a b E S such that 0 5 ab E S.

PROOF

Suppose that S is large in R. Let 0 # a E R and let < a >= {sa : s E S} be

the S-submodule of R generated by a. Then < a > nS 0 0, so there is a b E S

such that 0 : ab E S. Conversly, if 0 : I is an S-submodule of R, then there is an

0 $ a E I C R. By the hypothesis, there is a b E S such that 0 : ab E S. Since I is

an S-submodule, ab E I.
QED


Recall now from Chapter One that for an i-group G, C(G) is the collection of

i-ideals of G, and that C(G) is a complete, distributive, brouwerian sublattice of the

lattice of all subgroups of G.

The next characterization of the o-large f-subrings of an f-ring will depend on a

certain meet subsemi-lattice of C(G). We first need to set up some machinery.


Definition 3.1.3 Let G be an i-group. For any subset T C G, let T' = {a G :

Ial A It| = 0 Vt E T}. T' is called the polar of T in G.

We will use the following notational conventions. For g E G, g' = {g}1 and for

any subset T C G, T- = -{TI-}. g" is called the principal polar of g.


Definition 3.1.4 Let G be an i-group. P E C(G) is said to be a polar of G if P = T'

for some subset T C G.


If we consider I as a unary operation on C(G) then the following results are well

known and can be found in particular in Chapter 1 of "Lattice-Ordered Groups" [2].











1. BC B .

2. If B cC, then C CB .

3. B' = BI.

4. B' n C- = (B V C)' where V is the supremum in the lattice C(G).

5. P E C(G) is a polar if and only if P = P".

6. For any subset T C G, T' E C(G).

Let P(G) = {P E C(G) : P" = P}. The following is Theorem 1.2.5 [2]. That

P(G) is a complete boolean algebra is originally due to F. Sik [31] (1960). The

mapping context is part of a more general lattice theoretic result due to V. Glivenko

[18] (1929).

Theorem 3.1.2 P(G) is a complete boolean algebra when equipped with the meet n,

the join U defined by B U C = (B V C)" and complementation I. Furthermore the

map B B- B" is a lattice epimorphism from C(G) to P(G).

With this, the characterization we need is the following generalization of Theorem

11.1.15 [5]. These results are originally due to P. Conrad [7].


Theorem 3.1.3 Let G be an e-subgroup of the -group H. Then each of the following

implies the next.

1. G is o-large in H.

2. For every 0 < h H, there is a 0 < g G and an n N such that g < nh.


3. For each nonzero P E P(H), P n G 54 0.











4. For each 0 5 h E H there exists a 0 : g E h" f G.

5. The map P -+ P n G is a boolean isomorphism from P(H) to P(G).

Furthermore, 1 and 2 are always equivalent; 3,4 and 5 are always equivalent, and if

H is archimedean then 3 implies 1.

Now that we have some useful characterizations of essential and o-essential ex-

tensions we need to look at some particular extensions in both contexts. We will

first deal with the o-essential extensions. The following depends only on the e-group

structure.

For an -group G we will be interested in several o-extensions of G; the lat-

eral completion, the orthocompletion and, in the case where G is archimedean, the

Dedekind completion and the o-essential closure. We need the following definitions

and notation.

Definition 3.1.5 Let G be an e-subgroup of the I-group H. G is dense in H if for

every 0 < h E H, there is a 0
Clearly G is dense in H implies that G is o-large in H.

Definition 3.1.6 An t-group is Dedekind complete if every collection of elements

which is bounded above has a supremum. For G archimedean, an I-group G^ is

a Dedekind completion of G if G is an I-subgroup of G^, G^ is Dedekind complete

and each element of G" is the supremum of elements of G.

Definition 3.1.7 An I-group is laterally complete if every collection of pairwise dis-

joint elements has a supremum. An I-group GL is the lateral completion of G if G

is a dense i-subgroup of GL, GL is laterally complete and no proper i-subgroup of GL

contains G and is laterally complete.











Definition 3.1.8 Let G be an i-group. H E C(G) is said to be a cardinal summand

of G if there is a K E C(G) with H A K = 0 and H V K = G. We denote this by

G=H K.

Cardinal summands, in addition to being direct summands are unique. Also, if

G = H 1 K then both H and K are g-groups, and if g = h + k with h E H, k E K,

then 0 < g if and only if 0 < h and 0 < k.

Definition 3.1.9 An l-group G is called strongly projectable if for each P E P(G),

G = P P'. G is called projectable if for each g E G, G = g'- l g".

Definition 3.1.10 An l-group is orthocomplete if it is laterally complete and pro-

jectable. An l-group G is an orthocompletion of G if G is dense in Go, Go is

orthocomplete and no proper l-subgroup of Go contains G and is orthocomplete.

Definition 3.1.11 An l-group is o-essentially closed if it admits no proper o-essential

extensions. An o-essential closure of an l-group is an o-essentially closed o-essential

extension.

It should be pointed out that unless we restrict our attention to archimedean e-

groups an l-group always admits a proper o-essential extension because any g-group

can be lexicographically extended and will be large in this extension. For this reason,

the idea of an o-essential closure only makes sense in the category of archimedean

f-groups. The lateral completion of an l-group is always unique, if in addition G is

an archimedean l-group then G^ and G are unique as well.

Recall, that for an l-group G, P(G) is a complete boolean algebra. Let X =

St(P(G)) be the Stone dual of P(G). Since P(G) is complete, X is compact, Haus-

dorff and extremally disconnected. Recall that

D(X) = {f : X -- RU {oo} : f is continuous and f-(IR) is dense open}










For f,g E D(X), Y = f-'(R) fng-'(R) is a dense subset of X. Since X is extremally

disconnected, Y is C* embedded in X. Therefore fY = fX = X. We have f + g :
Y -- IR U {oo} with R U {do} compact. Therefore there exists a unique extension

of f +g to /Y = X. We then define the group element f +g to be this extension. We

then have that, for X extremally disconnected, D(X) with the pointwise ordering is
an i-group. We can also define pointwise multiplication as above making D(X) a

ring. In fact, D(X) is a complete vector lattice. The following theorem is due to S.

Bernau [4]. The earlier work of Pinsker and Vulich [32] obtains a similar result for
complete vector lattices.

Theorem 3.1.4 If G is an archimedean i-group then there is an t-isomorphism 7r of
G onto a large i-subgroup of D(X) that preserves all joins and meets. If G is a
vector lattice, 7r also preserves scalar multiplication. Furthermore, if {ex : E A}

is a maximal disjoint subset of G, r can be chosen so that {7r(ex) : A A} is a
set of characteristic functions of a family of pairwise disjoint clopen subsets of X

whose union is dense. In particular, if e is a weak order unit, then 7r can be chosen

so that 7r(e) is the characteristic function of X and if e is a strong order unit, then

r(G) C C(X). Finally if rli and r2 are any two such embeddings, then there is a
d E D(X) such that drl(g) = 72(g) for all g E G.

The following is Theorem 3.4 [7] and it provides us with an "umbrella" extension.

Theorem 3.1.5 Each archimedean i-group G admits a unique o-essential closure G".

G is the Dedekind complete, laterally complete, divisible i-group in which G is o-large
and Ge is i-isomorphic to D(X) where X is the Stone dual of the boolean algebra of

polars of G.










We now turn our attention to the ring theoretic aspects of essential extensions.
Recall that the classical ring of quotients of A is qA = S-1A, where S is the set of
regular elements of A. We need the following concept which is due to Y. Utumi [30].

Definition 3.1.12 Suppose that A is a subring of the ring B. B is called a ring of
quotients of A if for every bl, b2 E B, with b2 : 0, there is an a E A such that abl E A
and ab2 5 0.

With this definition, it is clear that qA is a ring of quotients of A, and that any
ring of quotients of A is an essential extension of A. It is not necessarily the case
that qA is a maximal essential extension of A; the maximal object is the so-called
complete ring of quotients, Q(A). What follows are two quite different constructions
of Q(A). The first is due to J. Lambek and can be found in Chapter 2 [22].

Definition 3.1.13 An ideal I in a commutative ring A is said to be dense if for every
a E A, al = 0 implies a = 0.

Let A be a commutative ring and let Hom(D, A) be the set of A homomorphisms
of D into A. Let F(A) = {f E Hom(D,A) : D is a dense ideal of A}. We call

f E F(A) a fraction. We can define addition and multiplication on F(A) as follows.
For fi e Hom(Di, A), f2 e Hom(D2, A),

1. fi + f2 E Hom(D n D2, A) is given by (fi + f2)(d) = f,(d) + f2(d).

2. fif2 E Hom((f2-1(D1)) n D1, A) is given by f1f2(d) = fi(f2(d)).

F(A) is closed under the above operations since for D1, D2 dense, both D1 n D2
and f2-1(Di) n D are dense. Now define an equivalence relation on F(A) by f, = f2

if fi = f2 on some dense ideal D. For f e F(A) let [f] denote the equivalence
class of f. Let Q(A) = {[f] : f E F(A)}. For a E A, let fa be multiplication










by a. If [fa] = 0 then aD = 0 for some dense ideal D, so that a = 0. Therefore

a F-+ fa is an monomorphism of A in Q(A). This mapping is called the canonical
monomorphism. With the above operations suitably altered for equivalence classes

we have the following theorem which is Proposition 1, 2.3 [22].

Theorem 3.1.6 If A is a commutative ring, then Q(A) is also a commutative ring. It
extends A and will be called its complete ring of quotients.

The following useful results can also be found in Lambek [22].

Theorem 3.1.7 [Prop.6 2.3[22]] Let A be a subring of the commutative ring B. Then

the following are equivalent.

1. B is a ring of quotients of A (in the sense of Utumi).

2. For all 0 f b E B, b-lA is a dense ideal of A and b(b-'A) 0.

3. There exists a monomorphism of B into Q(A) that extends the canonical monomor-

phism a F- fa of A into Q(A).

Recall that an R-module M is said to be injective if A and B are R-modules and

S: A -4 B is a monomorphism, then for any homomorphism 0 : A --+ M there is

a t : B -- M such that c o a = 4. R is called self-injective if it is injective as an

R-module. The following are all found in Chapter 4 [22].

Lemma 3.1.1 Q(Q(A)) = Q(A).

If we restrict ourselves to the case where A is semi-prime, we have the following

theorems.


Lemma 3.1.2 If A is semi-prime, then Q(A) is self-injective.










Lemma 3.1.3 If A is semi-prime, then Q(A) is the injective hull of A.

Equivalently, Q(A) is the maximal essential extension of A. This gives us the
following lemma.

Lemma 3.1.4 If A is an essential extension of B, then Q(A) is an essential extension

of Q(B).

We now look at Banaschewski's construction of Q(A) [3]. This is a generalization
of the construction carried out in "Rings of Quotients of Rings of Functions" [12]
where it is shown that if X is a Hausdorff space, then

Q(X)= Q(C(X))= lim{C(U) : U C X is dense, open}

Let A be a commutative semi-prime ring with identity. Let f C Spec(A) be such
that n Q = 0. Call such a family of prime ideals a separating family. Topologize Q
with the hull-kernel topology. Let D = {U C Q : U is dense, open}. For U E D, let
Au = {f E IpEu q(A/P) : for every Q E U there is a neighborhood V with Q E
V C U and there are a,b E A with f(P') = (a + P')/(b+ P') for every P' E V}. For
each U E Q, dense, open, Au is a subring of I1Pe q(A/P) containing A. If U C V
are dense, open in Q, then the map rvu : Av -- Au given by irv,u(f) = flu is
a ring homomorphism. It can then be shown that the rings Au together with the
maps 7v,u form a directed system indexed by I= {U E D} where V < U if U C V.
Banaschewski's result is the following

Theorem 3.1.8 Let A be a commutative semi-prime ring, and Q a separating family
of primes. Then,


Q(A) = lim(Au, rv,u)










In particular, since A is semi-prime, if we take Q = Min(A), then the canonical

map A F- A/P -+ q(A/P) induces an imbedding

A -+ A/P i q(A/P)
PEMin(A) PEMin(A)
Moreover, in the case that A is an f-ring, since each minimal prime ideal is an i-
ideal, each A/P is an ordered ring and hence each q(A/P) is an ordered field and
the canonical maps are all f-homomorphisms. If we take the pointwise ordering on

HPEMin(A) A/P and HPEMin(A) q(A/P), then these too are f-rings and the induced
maps are f-homomorphisms. That a ring A has a unique maximal essential exten-
sion was first shown by Y. Utumi [30]. That there is a canonical ordering on Q(A),
extending the ordering on A, for which Q(A) is an f-ring was shown by F. Anderson
[1]. The following theorem bringing together these results with Banaschewski's con-
struction of Q(A) is due to J. Martinez [24]. Recall that a ring A is a Von Neumann
regular ring if for every a E A there is an x E A such that axa = a.

Theorem 3.1.9 For each semiprime f-ring A, the maximal ring of quotients Q(A)
admits a lattice-ordering making it a semi-prime f-ring and containing A as an f-
subring. f E Q(A) is positive if for each dense open subset W of Min(A), and
each P E W there exists an open set U and positive elements a and b in A, so that

P E U C W and for each Q E U, f(Q) = (a + Q)/(b + Q). This is the unique
lattice-ordering on Q(A) making it an f-ring containing A as an f-subring. Finally,

Q(A) is a Von Neumann regular ring.

3.2 Quasi-Specker f-Rings and f-Rings of Specker-Type

Let A be a commutative semi-prime f-ring with identy and bounded inversion.
Recall that S(A) is the Q subalgebra of A generated by the idempotents of A, and











that if A is an f-algebra, Si(A) is the real subalgebra generated by the idempotents

of A.

Definition 3.2.1 We say that A is a specker-type ring if S(A) is a large subring of

A. If A is an f-algebra such that SR(A) is a large subring, then A is a specker-type

algebra. We say that A is a quasi-specker ring if S(A) is o-large in A.

When A is an f-algebra, the distinction between being a specker-type ring and a

specker-type algebra is unimportant in terms of the method and content of the proofs

that follow. The significant difference is that the objects which characterize the two

specker properties are possibly different. We don't need to make this distinction for

the definition of quasi-specker, since in the case that A is an f-algebra, S(A) is order

large in Sn(A). Since many of the results are derived for the special case A = C(X)

we will call a Tychonoff space X a specker space or a quasi-specker space if C(X) is

a specker-type algebra or a quasi-specker ring respectively.

We will need the following definition.

Definition 3.2.2 A commutative semi-prime ring A is locally inversion closed if for

every a E A, and every P E Min(A) with a ( P, there is an open neighborhood U,

with P E U C fa, and there exists b E A such that ab = 1 mod Q for every Q E U.


The following characterization is due to J. Martinez [24].

Theorem 3.2.1 A semi-prime f-ring A is locally inversion closed if and only if for

each 0 < a E A, there exists {c : i E I} such that 0 < ci < a and axi = ci for some

xi E Ac, and a1 = nci'.

Theorem 3.2.2 Let A be a commutative semi-prime f-ring with 1 and bounded inver-

sion. If A is local-global and J(A) = 0 then A is locally inversion closed.










PROOF

Let 0 < a E A. Then Ma is a nonempty open subset of Max(A). Since A is
local-global, by Theorem 2.3.1, Max(A) is zero-dimensional and so has a base of

clopen sets. Let M E Ma. Then there is a clopen KM C Max(A) with M E KM C

Ma. Since KM is clopen, by Lemma 2.3.6, there is an eM E A idempotent with

KM = AeM. We claim that eM E a"1; otherwise, there is 0 < x E A with x A a = 0
and x A eM 7 0. Since J(A) = 0, there is an N E Max(A) with x A eM N. Since

N is an 1-ideal, we have that x 0 N and eM N. Then N E Mem C Ma, so that

a 0 N. Since x A a = 0 and N is a prime 1-ideal, x E N; a contradiction. Therefore

eM E a". In particular, eM A a 7 0.

Since eM is idempotent we can write A as A = AeM ED A(1 eM). Considering

AeM we have that this is both an ideal and an e-ideal of A. Moreover, any ideal of

AeM is an ideal of A.
Now let P E Max(AeM). Since eM V P, the ideal generated in A by P and

1 eM is proper and so is contained in some N E Max(A) with eM V N. Then

N C MeM C Ma so that a V N as well. Therefore aeM 0 N as N is a prime ideal,

and P C N so that aeM V P. We have shown that aeM misses every maximal ideal

in Max(AeM) so that aeM is a multiplicative unit in AeM. Therefore there is an

XM E AeM such that aOfc IA = eM, the multiplicative identity in AeM.

Now for each M E M,, let eM and XM be as above and let CM = aeM We have

1. 0 < CM < a. This holds since 0 < eM 1 and eM A a : 0.

2. a(aeMXM) = aeM = cM and aeMXM = CMXM E ACM.

It remains to show that a1 = cM,. Clearly ac C fcM' as CM is a multiple

of a. Suppose 0 < b a a'. Then b A a 0. Since J(A) = 0, there is a maximal

ideal M with b A a 4 M. By convexity, b A M and a V M. Therefore M E Ma. As











shown above, there is an idempotent eM with M E MeM C M,. Let cM = aeM. If

bA cM = 0, since b M and M is prime, CM E M. Then CMeM = a(eM2) = aeM c M

so that a M or eM e M. This is a contradiction since M E MeM C MA. Therefore

bA CM A 0 and so b f cM-
QED



Lemma 3.2.1 Let A be a commutative semi-prime f-ring with 1 and bounded inver-

sion. Then S(A)L = Q(S(A)). If A is an f-algebra, then S(A)L = Q(Sg(A))


PROOF

Since S(A) is an archimedean f-ring with bounded inversion, by Theorem 1.8

[24], Q(S(A)) = S(A)o the orthocompletion of S(A). Since S(A) is archimedean,
S(A)o = S(A)L (Theorem 8.2.5 [2]). The proof is identical if A is a vector lattice.

QED

It is in this context of lateral completions that the differences between S(A) and

SR(A) first becomes apparent as the following example indicates.

Example Let A be the ring of eventually constant real valued sequences. Then A

is an f-algebra. The idempotents of A consist of {0, 1} sequences that are eventually

constant. Then S(A) consists of the eventually constant rational sequences and

S(A)L = I Q, whereas, SR(A) consists of the eventually constant real sequences and

Sa(A)L = FR.
The first main result is the following theorem which characterizes specker-type

f-rings in terms of the complete ring of quotients. An analogous result holds for

f-algebras.


Theorem 3.2.3 Let A be a commutative semi-prime f-ring with 1 and bounded inver-

sion. Then the following are equivalent.










1. A is an specker-type ring.

2. A is a subring of Q(S(A)).

3. S(A)L = Q(A).

PROOF

That (1) implies (2) follows directly from the definition since Q(S(A)) is the
maximal S(A)-essential extension of S(A). If S(A) C A C Q(S(A)) then A is an

S(A)-essential extension of S(A) so that (2) implies (1). We will now show that
(2) implies (3). Suppose that A C Q(S(A)). For 0 a E Q(S(A)), by Theorem

3.1.1 there is a 0 6 b E S(A) such that 0 5 ab E S(A). Since S(A) C A, Q(S(A))

is an A-essential extension of A. By Lemma 3.1.1 and Theorem 3.1.8, Q(A) C

Q(Q(S(A))) = Q(S(A)). Since A is a subring of Q(S(A)), A is an S(A)-essential
extension of S(A) so that Q(S(A)) C Q(A). Therefore Q(A) = Q(S(A)). By Lemma
3.2.1, S(A)L = Q(A). Since A is a subring of Q(A), that (3) implies (2) also follows
directly from Lemma 3.2.1.
QED


We will now consider how specker-type rings, quasi-specker rings and local-global
rings relate to one another. Recall that for any commutative semi-prime f-ring with
identity and bounded inversion, J(A) C {x E A : |Ix << 1}, and therefore if A is

archimedean then J(A) = 0.

Corollary 3.2.1 Let A be a commutative semi-prime f-ring with identity and bounded

inversion. If A is an specker-type ring, then A is archimedean. The same conclusion
obtains for f-algebras.











PROOF

Since A is a specker-type ring, A is an f-subring of S(A)L. Since S(A) is archimedean,

by Theorem 8.2.5 [2] its lateral completion S(A)L is as well. An f-subring of an

archimedean f-ring is archimedean, so that A is archimedean.
QED



Theorem 3.2.4 Let A be a commutative semi-prime f-ring with identity and bounded

inversion. If A is a specker-type ring then A is a quasi-specker ring. The same

conclusion obtains for f-algebras.


PROOF

Let P E 'P(A). Since A is an f-ring, P is a ring ideal and therefore an S(A)

submodule of A. Since A is a specker-type ring, P n S(A) 4 0. Therefore, by

Theorem 3.1.3, A is a quasi-specker ring.
QED



Theorem 3.2.5 Let A be an archimedean commutative semi-prime f-ring with identity

and bounded inversion. If A is a local-global ring, then A is a quasi-specker ring.


PROOF

Let P e P(A) and let 0 7 a E P. As in the proof of Theorem 3.2.2, there is an

e E A idempotent with e E a". Since a E P, a" C P, so that e E P n S(A) and A

is a quasi-specker ring.
QED


We will eventually see examples in the case A = C(X) to show that none of the

two previous implications reverse.










3.3 Quasi-Specker Spaces and Specker Spaces

In this section we will apply the preceding results for the case A = C(X). Follow-
ing the exposition in Rings of Continuous Functions by Gillman and Jerison [15], we
consider for a space X, the ring of continuous real valued functions C(X). All spaces
will be assumed to be Tychonoff. We recall the following notation. For f E C(X),

coz(f) = {x E X : f(x) # 0} and Z(f) = { E X : f(x) = 0}. Some observations
and recollections are in order.

1. For X Tychonoff, {coz(f) : f E C(X)} is a base for the open sets of X.

2. If we define a partial ordering on C(X) by f 2 0 if f(x) > 0 for all x E X,
and operations pointwise, then C(X) is a commutative semi-prime f-ring with
bounded inversion and identity.

We will use the following notational conventions. Q(X) = Q(C(X)), q(X) =

q(C(X)), S(X) = SK(C(X)), C*(X) = {f E C(X) : f < n for some n E N}.

Lemma 3.3.1 Let X be a Tychonoff space. Then C(6X) is -isomorphic to C*(X)

and S(#X) is f-isomorphic to S(X).

PROOF

That C(3X) and C*(X) are isomorphic as rings is Theorem 4.6(i) in [27] and is
given by f i-f fO from C*(X) to C(3X). That this map preserves order follows from
the density of X in /3X. The second e-isomorphism follows from the first and the
observation that S(C*(X)) = S(X).
QED

In subsequent arguments involving C(PX) and S(P3X), we will use the above
e-isomorphisms extensively; that is, to prove a result for C(PX), we will prove the










result for C*(X). The next theorem is an improvement of Theorem 3.2.3 for the case
A = C(X)

Theorem 3.3.1 Let X be a Tychonoff space. The following are equivalent.

1. X is a specker space.

2. For every 0 $ f E C(X) there is a K C X clopen and a 0 ~ c E R such that

fIK=C.

3. 3X is a specker space.

4. S(X) = Q(X).

PROOF
Suppose that X is a specker space. Let 0 = f E C(X). Since X is a specker space,
there is an s E S(X) such that 0 6 sf E S(X). We can write s = rIXK, + +rnXKn
where we may assume that the ri are distinct, nonzero and the K, are disjoint, clopen.
Since 0 f sf E S(X), we can write sf = q1XKI + .. + qmXK' where again, the qi
are distinct, nonzero, and the Kf are disjoint, clopen. Then for any 1 < i < m,

sf K, = qi 0. In particular, sIK, 0. Then K' C coz(s) so that K'I n Kj 5 0 for
some 1 < j < n. Let T = K' n Kj. Then T is clopen and sf T = rjfIT = qi, so that

fIT = qi/rj = 0.
Suppose now that (2) holds. Let 0 7 f E C(X). Then there is a 0 = K C X
clopen and a 0 $ c E R with f|K = c. Let s = XK. Then s E S(X) and 0 0 sf E
S(X). Therefore X is a specker space.
We will next show that (2) and (3) are equivalent. Suppose that (2) holds. It
suffices to show that C*(X) is a specker-type ring. Let 0 7 f E C*(X). Since
C*(X) C C(X), there is a K C X clopen and a 0 7 c E R such that fIK = c. Let

s = XK. Then s E S(C*(X)) = S(X) and 0 $ sf E S(X).










Now suppose that #X is a specker space. Let 0 : f E C(X). We will first show
that we may assume that 0 < f. If 0 4 f then either f < 0 or coz(f V 0) 5 0. If

f < 0 then -f > 0 and -f IK = c / 0 implies fIK = -c # 0. If coz(f V 0) # 0 then
there is a 0 5 T C X with T C coz(f V 0). Then 0 < g = f IT C(X). If there is a

0 # K C X clopen and a c 5 0 with gIK = c, then glK = fITngK = c with 0 : T n K
clopen.
We will show that if C*(X) is a specker-type ring then (2) holds. If f < 1,
then f E C*(X) and we are done. Assume then that f X 1. Then (f V 1) > 1
and (f V 1) $ 1. Since C(X) has bounded inversion, (f V 1)-1 E C(X) and 0 <

1 (f V 1)-1 < 1 with (f V 1)-1 # 0,1. In particular, 1 (f V 1)-1 E C*(X). Then
there is a 0 K C C X clopen, and a c 0 0 with 1 (f V 1)-1IK = c, where 0 < c < 1.
Now, 0 < (f V 1)1K = 1 c < 1 so that (f V 1)IK = 1- > 1. Therefore, fl > 1IK
so that fIK = (f V 1)|K = C 0.
The equivalence of (1) and (4) follows directly from Theorem 3.2.3, taking A =
C(X).
QED

Since a subset of X is clopen if and only if its characteristic function is continuous,
it is straightforward to show that if X is a (quasi-)specker space and Y C X is clopen,
then Y is a (quasi-)specker space. As immediate corollaries of Theorem 3.3.1, we have
the following results.

Corollary 3.3.1 If X contains a dense set of isolated points then X is a specker space.

PROOF
If 0 $ f E C(X), then by density, there is an isolated point x with f(x) : 0.
Take K = {x} and apply (2) in Theorem 3.3.1.
QED











Definition 3.3.1 A point p in X is called a p-point if p E Z(f) implies that p E

int(Z(f)).

Corollary 3.3.2 If X is zero-dimensional and contains a dense set of p-points, then

X is a speaker space.

PROOF

We will again appeal to (2) in Theorem 3.3.1. Suppose that 0 # f E C(X). By

density, there is a p-point p such that f(p) = c 7 0. Since p is a p-point, there is an

open set O with p E O such that flo = c. Since X is zero-dimensional, there is a K

clopen with p E K C O. Then f K = c and therefore X is a specker space.
QED


Definition 3.3.2 A collection B of open sets of X is called a r-base if for every non-

empty open set 0 there is a 0 # U E B with U C O.

We have the following near analogue of Theorem 3.3.1 for quasi-specker spaces.

Theorem 3.3.2 Let X be a Tychonoff space.

1. X is a quasi-specker space.

2. /3X is a quasi-specker space.

3. X has a clopen i-base.

Then (1) and (2) are equivalent; (1) implies (3), and if X is compact then (3) implies

(1).

PROOF

Suppose that X is a quasi-specker space. Then S(X) is order large in C(X). Since

S(X) C C*(X) C C(X) as f-rings, S(X) is order large in C*(X). Suppose now that










PX is a quasi-specker space. Let 0 <_ f C(X). Then 0 < (f A 1) e C*(X), so there

is an s E S(X) and an n E N such that 0 < s < n(fA 1). But n(fA1) = nf An < nf

so that X is a quasi-specker space.

To show that (1) implies (3), let 0 $ O C X be open. Since X is Tychonoff, there

is an f E C(X) with 0 : coz(f) C O. By the hypothesis, there is an 0 # s E S(X)

and an n E N such that 0 < s < nf. Say s = r1XK1 + X+ rXK where as before

the rid's are distinct, nonzero and the Ki's are nonempty disjoint clopen subsets of X.

Then for any i, Ki C coz(nf) = coz(f) C O. Therefore X has a clopen 7r-base.

Suppose now that X is compact with a clopen 7-base. Let 0 < f E C(X) with

0 f f. Then there is a 0 5 K clopen with K C coz(f). Since X is compact and K

is clopen, K is compact. Therefore there is an 0 < a E R such that a = min{f(x) :

x E K}. Choose n E N so that n > 1. If we take s = XK E S(X), then nf > if > s
and X is a quasi-specker space.
QED

We have seen in the previous section that for f-rings (respectively f-algebras) if

A is a specker-type ring (algebra) then A is a quasi-specker ring, and if A is a local-
global ring with zero Jacobson radical, then A is a specker-type ring (algebra). In

the present context these results translate as follows.

First, by taking A = C(X) in Theorem 3.2.4, we get the following corollary.

Corollary 3.3.3 If X is a speaker space then X is a quasi-specker space.


Corollary 3.3.4 If X is strongly zero-dimensional then X is a quasi-specker space.

PROOF

If X is strongly zero-dimensional then fX is compact and zero-dimensional.

C(fX) is then local-global, and as C(Y) is archimedean for any space Y, #X is










a quasi-specker space, by Theorem 3.2.5. Therefore, by Theorem 3.3.2, X is a quasi-

specker space.
QED


The following examples show that the converse of the above corollaries do not

hold and that being a specker space is independent of strong zero-dimensionality.

Example Of a space X which is a specker space and hence a quasi-specker space,

but which is not strongly zero-dimensional.

Let N be the natural numbers with the discrete topology and let aN denote the

one point compactification of N where we denote by oo the point adjoined. Let

X = [0, 1] x aN where the topology is a refinement of the product topology such that

points of the form {(r,n)} for n E N are isolated.

The idea is that the isolated points of aN remain isolated. Suppose now that

0 I f E C(X). Then f(r,n) = c 0 for some (r,n) E X \ {[0, 1] x {oo}}. Since

(r,n) is isolated, s = X{(r,n)} E S(X) and 0 5 cx{(r,n)} = sf E S(X). Therefore X
is a specker space. Since X contains [0, 1] as a connected subset, it is not (strongly)

zero-dimensional.

Example Of a space X which is strongly zero-dimensional and hence a quasi-

specker space, but is not a specker space.

Let X = Q as a subspace of R with the topology generated by the open intervals.

Then X is strongly zero-dimensional; intervals of the form (r, s) with r, s E IR\ Q are

a clopen base. Let f E C(X) be the identity map. Suppose there is an s E S(X) such

that 0 7 sf E S(X). Then write sf = rlXKi +' + rnXKn where the rid's are distinct

nonzero and the Ki's are nonempty disjoint clopen and s = qlXK' + + qmXK,

similarly. Then there are i,j such that T = KinKK 5 0. T is clopen and f T = ri/qj.

This is a contradiction since points are not clopen.










These examples show that the following hold.

1. That "X is a specker space" is independent of whether or not X is strongly

zero-dimensional.

2. That X is a quasi-specker space does not imply that X is a specker space.

3. That X is a quasi-specker space does not imply that X is strongly zero-

dimensional.

3.4 The Absolute of a Hausdorff Space

In what follows we will consider one construction of the absolute of a Hausdorff

space and some of its properties. The following is the construction presented in

"Extensions and Absolutes of Hausdorff Spaces" [27], Chapter 6.6 of the Gleason

Absolute of a compact Hausdorff space first done by A.M. Gleason [17].

We will need the following definitions.

Definition 3.4.1 Let X and Y be topological spaces and let f : X Y be a function.

Then f is said to be perfect if f is closed and for each y E Y, f-l(y) is a compact

subset of X.

In the definition of perfect maps there is no assumption about continuity, but in

the case that f is continuous and X is compact, f is perfect.

Definition 3.4.2 Let X and Y be topological spaces and let f : X Y be a function.

Then f is said to be irreducible if f is closed, onto, and if K is a proper closed subset

of Y, then f(K) 5 Y.

Irreducible maps have many interesting and useful properties. For more on irre-

ducible maps, see Chapter 6.5 [27]. In particular, irreducible maps have the following

properties.










Lemma 3.4.1 Let X, Y, and Z be spaces, f : X -- Y an irreducible map. Then,

1. If g : Y -- Z is an irreducible map then so is the composite g o f : X -+ Z.

2. If0 $ U c X is open, then int(f(U)) 0.

3. If W C Y is dense, then f-'(W) is dense and ff-,(w) f-'(W) W is

irreducible.


Example Let aN denote the one point compactification of the naturals. The map

from : ON -* aN given by 0(n) = n for n E N and (P/N \ N) = oo is irreducible.

Moreover, if N is any compactification of N then the Stone extension iZ : ON -- N of

the embedding of N in N is irreducible.


Definition 3.4.3 A space X is said to be extremally disconnected if the closure of

each open set is open.

If we restrict our attention to Tychonoff spaces we have the following, which is a

compilation of results from [27], 6.2 Theorems (b) and (c), and [15] problem 3N4.


Theorem 3.4.1 Let X be a Tychonoff space. The following are equivalent.

1. X is extremally disconnected.

2. Disjoint open sets in X have disjoint closures.

3. Each dense (open) subset of X is extremally disconnected.

4. Each dense (open) subset of X is C* embedded.

5. fX is extremally disconnected.

6. C(X) is Dedekind complete.










Example Clearly any discrete space D is extremally disconnected. Theorem
3.4.1 tells us then that PD is also extremally disconnected. Both D and fD contain
a dense set of isolated points. To find an extremally disconnected space without
isolated points, begin with a complete atomless boolean algebra B. Then St(B) is a
compact extremally disconnected space without isolated points.

Definition 3.4.4 For a space X, a subset K of X is called a regular closed subset if
K = cl(int(K))

Now let X be a compact Hausdorff space and let RI(X) denote the regular closed
subsets of X. We define the following operations on R(X).

Definition 3.4.5 For A, B E R(X), B C 7(X), define

1. A binary meet by A A B = cl(int(A n B)).

2. A binary join by AVB = AU B.

3. Complementation by A' = X \ int(A).

and an infinitary meet and join by

4. AB = cl(int(nB)) and VB = cl(int(UB))

By Proposition 2.2(c) and Example 3.1(e)(4) in [27], (R(X), A, V,') is a complete
Boolean algebra. Let EX denote the Stone dual of R(X). Then EX is a compact,
Hausdorff extremally disconnected space. For emphasis we will recall here that the
points of EX are RZ(X)-ultrafilters and that the basic open sets of EX are of the
form: for K e R(X), OK = {a EX : K al. By Theorem 6.6(e) in [27] the map
e: EX X, defined by e(a) = na, is a continuous irreducible surjection.










The above construction generalizes to the non-compact case by taking EX to be

the subspace of the Stone dual of R7(X) whose points are the fixed ultrafilters. In this

case, EX is an extremally disconnected zero-dimensional dense subspace of the Stone

dual of RI(X) and e : EX -- X as defined above is a continuous perfect irreducible

surjection. We will call e the canonical surjection.

The space EX as defined above is called the Gleason Absolute of X, in the case

that X is compact, and the Illiadis Absolute for X non-compact. We will agree to

refer to EX as the absolute of X in either case. In addition to the above mentioned

properties of the absolute, EX is unique in the following sense.


Theorem 3.4.2 Let X be a Hausdorff space and let (Y, ) be such that Y is a zero-

dimensional, extremally disconnected space and i : Y -* X is a perfect irreducible

function. Then there is a homeomorphism 0 : EX -+ Y such that f o 0 = e.

Of particular interest will be the relationship between o-large embeddings of f-

rings and irreducible maps of topological spaces. Recall that if A is a commutative

semi-prime f-ring with identity and bounded inversion and if B is an f-subring of A

with bounded inversion and 1 E B, then there is an induced map : Max(A) -+

Max(B) that is a continuous surjection.

We translate part of the exposition contained in Chapter 11 of "Rings of Quotients

of Rings of Functions" [12] into a language suitable for our purposes. The discussion

that follows was originally done for annihilator ideals of semi-prime rings and the

Boolean algebra of regular open sets. Let P e P(A) and consider Kp = {M E

Max(A) : M D P} and Op = {M E Max(A) : M P'}. Then Kp is closed,

Op is open and since maximal ideals are prime, Op C Kp. It is always the case

that cl(Op) = Kp and if J(A) = 0, then int(Kp) = Op. In particular if J(A) = 0,

then Kp is a regular closed set. Moreover, when J(A) = 0 the map P H- Kp is a










Boolean isomorphism from P(A) onto TR(Max(A)) (Theorem 11.8 [12]). Therefore,
if J(A) = 0, the regular closed sets of Max(A) are precisely those of the form
Kp = {M E Max(A): M D P} for some P E P(A).

Theorem 3.4.3 Let A, B and q be as in the preceding paragraphs. If B is o-large in
A, then 0 is irreducible. If in addition A is archimedean then the converse holds.

PROOF
Since Max(A) is compact and Max(B) is Hausdorff, q is closed. Recall that
is defined by O(M) = the unique maximal ideal of B containing M n B. Denote by
A/. the basic open sets of Max(B) and by Ma the basic open sets of Max(A). Let
K be a proper closed subset of Max(A). Then there is an 0 < a E A such that
M, C Max(A) \ K. Since B is o-large, there is a b E B and a n E N such that
0 < b < na. Then Mb C Ma so that K C Max(A) \ Mb. If M E Max(A) \ Mb,
then b E M and as b E B, b E Mn Max(B) C O(M). Then O(M) E Max(B) \ AF
and consequently O(K) C Max(B) \ A/b Max(B). Therefore is irreducible.
For the converse we will show a bit more. That if J(A) = 0 and q is irreducible
then the contraction P '-P P n B is a Boolean isomorphism from P(A) onto P(B).
Then by Theorem 3.1.3, in the case that A is archimedean, the result follows. Also
by Theorem 3.1.3, suffices to show that if 0 f a E A then a1l n B 0 0. Suppose
we know that this holds for B(1) C A(1). Let denote the polar operation in A(1).
Let 0 : a E A. Since A(1) is order large in A we have that a1 nf A(1) 5 0. In fact,
all n A(1) = (a A 1)**. Therefore (a A 1)** n B(1) # 0, so that a1 n B 5 0. Since
Max(A) 2 Max(A(1)), Max(B) Max(B(1)) and J(A) = 0 implies J(A(1)) = 0,
we may assume without loss of generality that A has strong unit.
Suppose that A has a strong unit with J(A) = 0, B is an f-subring of A and

S: Max(A) --, Max(B) is irreducible. Since A has strong unit, O(M) = M n B.










Let 0 $ a E A. Since J(A) = 0, there is an M E Max(A) with a V M, so that
K = {M E Max(A): M D a-} is a proper regular closed subset of Max(A). Since
is irreducible O(K) is a proper regular closed subset of Max(B). Therefore there
is a Q E P(B) such that b(K) = {N E Max(B): N D Q}. Each N E O(K) is of the
form N = Mn B for some M E K so that O(K)) = {M n B :M BD Q}.
Suppose now that a1 n B = 0 and Q # 0. Then there is a 0 < q E Q with

q V a". Since q V a"1, there is a 0 < c E a1 with 0 < cq. Since J(A) = 0, there
is an M E Max(A) with cq M. Then c V M and q V M. Since c M, a1 ( M
and as M is prime, a-1 C M. Then M E K so that O(M) = M n BD Q. This is
a contradiction as q V M. Therefore, if aI n B = 0 then Q = 0. If Q = 0 then
O(K) = Max(B), but is irreducible and K is a proper closed subset of Max(A) so
that Q $ 0, and hence a11 n B = 0.
We have then for 0 7 a E A that all nB $ 0 and so by Theorem 3.1.3, the map
P P n B is a Boolean isomorphism from P(A) onto P(B).
QED


We can also argue from A archimedean to the order largeness directly by noting
that if A is archimedean and 0 is irreducible, by Lemma 11.8 in [12], the map P F-+
{M E Max(A) : M D P} is a Boolean isomorphism from P(A) onto R(Max(A)).
Since B also is archimedean we have that P(B) is Boolean isomorphic to R(Max(B)).
By the embedding theorem for archimedean 1-groups, Be = Ae. Then B C A C B'
and since B is o-large in Be, it is o-large in A.

3.5 Specker Spaces and Absolutes

In this section we look at the "inheritability" of being a specker-type space in the
context of Tychonoff spaces vis-a-vis their absolutes.











Lemma 3.5.1 Suppose that A = S6(A) and AL = Ae. Then Max(A) is a specker

space.


PROOF

Since A = S(A), A is hyper-archimedean and therefore, Max(A) Min(A)

is zero-dimensional. Since Max(A) is compact, by Corollary 2.3.4, C(Max(A)) is

local-global. Then by Theorem 3.2.5, C(Max(A)) is a quasi-specker ring. That

is S(Max(A)) is o-large in C(Max(A)), so that C(Max(A)) is an f-subring of

S(Max(A))e.

For any ring R, let Id(R) denote the idempotents of R. For a E Id(A), the map

a -4 XMa extends linearly to an f-ring isomorphism from A onto S(Max(A)). There-

fore S(Max(A))e Ae = AL S(Max(A))L. By Lemma 3.2.1, S(Max(A))L =

Q(S(Max(A))) therefore C(Max(A)) C Q(S(Max(A))).
QED



Lemma 3.5.2 If X is a quasi-specker space and S(X)L is o-essentially closed then X

is a specker space.


PROOF

Since S(X) is a quasi-specker, S(X) is o-large in C(X). By Theorems 3.1.3 and

3.1.5, S(X)e = C(X)e. By Lemma 3.2.1, S(X)L = Q(S(X)), therefore C(X) C

Q(S(X)) and X is a specker space.
QED


In the preceding lemma we cannot omit the assumption that X is a quasi-specker

space, as the following example shows.

Example Let X = [0, 1] with the open interval topology. Since X is connected,

the only idempotents in C(X) are the constant functions 0 and 1. Then S(X) R











and is essentially closed, as is S(X)L, but X has no proper clopen subsets and is

therefore not a specker space.

Lemma 3.5.3 If X is a compact quasi-specker space and EX is a specker space, then

X is a speaker space.

PROOF

Let 0 : f E C(X). Then 0 7 f oe E C(EX), where e is the canonical surjection.

Since EX is a speaker space, there is a nonempty clopen K C EX and a c : 0 with

fo e (K) = c. Since K is clopen hence regular closed and e is irreducible, e (K) is

a nonempty regular closed subset of X. In particular, int(e (K)) $ 0. Since X is

compact and quasi-specker, there is a nonempty B C X clopen with B C int(e (K)).

Then f(B) = c so that X is a specker space.
QED


The preceding result generalizes to the following.

Theorem 3.5.1 If X is a quasi-specker space and EX is a specker space, then X is a

specker space.

PROOF

Since X is a quasi-specker space, fX is a compact quasi-specker space. Since

EX is a specker space, 3(EX) is a specker space. Since X is Tychonoff, by Theorem

6.9(b)(4) in [27], /(EX) E(OX) so that E(/X) is a specker space. By Lemma

3.5.3, fX is a specker space and therefore X is a specker space.
QED


For extremally disconnected spaces the following characterization obtains.

Theorem 3.5.2 If X is extremally disconnected then X is a specker space if and only

if S(X)L is essentially closed.










PROOF

Suppose that X is a specker space. Since X is extremally disconnected, by The-

orem 3.4.1, C(X) is Dedekind complete. Since X is a specker space, C(X) is an

o-large subring of S(X)L. Since X is Tychonoff and extremally disconnected, X is

strongly zero-dimensional and hence a quasi-specker space. In particular, S(X) is

o-large in C(X). Since C(X) is Dedekind complete, S(X)^ is an o-large subring of

C(X) and hence of S(X)L. But then S(X)e = (S(X)^) C S(X) C S(X)e and

S(X)L is essentially closed.

Suppose now that S(X)L is essentially closed. Since X is extremally disconnected,

X is a quasi-specker space and the result follows from Lemma 3.5.2.
QED


We have shown that for quasi-specker spaces, if the absolute is a specker space,

then the space itself is a specker space. We would like to reverse this implication. So

far we have only partial success.


Theorem 3.5.3 Let X be a quasi-specker space. If S(X)L is essentially closed then

EX is a specker space.


PROOF

The continuous map e : EX -- X induces an f-ring homomorphism C(e) :

C(X) -- C(EX) given by; for f E C(X), C(e)(f) = f o e. Since e is onto,

C(e) is an embedding and since e is perfect and irreducible, this embedding is o-

large. Then since X is a quasi-specker space, S(X) is o-large in C(X) and therefore

S(X) is o-large in C(EX). The restriction of C(e) to S(X) is an embedding into

S(EX), and as S(X) is o-large in C(EX), S(X) is o-large in S(EX). Therefore,










S(X)L C S(EX)L. By the hypothesis, S(X)L = S(X)e. Since S(X) is o-large in

C(EX), C(EX) C S(X){. Thus C(EX) C S(EX)L and EX is a specker space.
QED


Definition 3.5.1 A space X is said to have the countable chain condition if every

collection of disjoint open sets is at most countable.

The following conjecture is the motivation for much of what follows.


Conjecture 3.5.1 Let X be a compact Tychonoff space with the countable chain con-

dition. If X is a specker space, then EX is a specker space.


Recall that a collection of open subsets B of a space X is called a ir-base for X

if for every nonempty open 0 C X there is a nonempty U E B with U C O. As a

special case of this conjecture we will consider the case for X a compact Tychonoff

space with a countable r-base where we can get a positive result. We will then look

at what progress has been made with regards to the conjecture itself. We need the

following notion which is due to J. Martinez [23].


Definition 3.5.2 A Tychonoff space X is said to be a weakly specker space if whenever

0 0 f E C(X), then for each open set V on which f does not vanish, there exists an

open U C V such that flu is nonzero and constant.


Lemma 3.5.4 If X is weakly specker, then for any 0 = f E C(X), there is a collection

of open sets with dense union such that f is constant on each open set.


PROOF

Let S = {O, : flo, = co 5 0} U {O : flo, = 0}. If S is not dense, then

V = X \ cl(S) is a nonempty open set on which f does not vanish. Since X is











specker, there is an open U C V such that flu is nonzero and constant. This is a

contradiction. Therefore, S is dense.
QED


The following is an unpublished result due to W. W. Comfort.

Theorem 3.5.4 If X is a metric space with no isolated points, then there is a contin-

uous f : X -+ [0,1] such that if U is a nonempty open subset of X, then If(U)I > 2.

For our purposes, the importance of Comfort's Theorem is the following corollary.

Corollary 3.5.1 If X is a weakly specker metric space, then X contains a dense set

of isolated points.

PROOF

If X contains no isolated points, then the function guaranteed by Theorem 3.5.4

is not constant on any open set. Therefore a weakly specker metric space contains at

least one isolated point. Let D be the set of isolated points of X. If cl(D) : X, then

X\cl(D) is a nonempty open set without isolated points. Therefore K = cl(X\cl(D))

is a metric space that is a closed subspace of X that contains no isolated points. Let

0 7 f E C*(K) be the function guaranteed by Theorem 3.5.4. By Tietze's extension
theorem, there is an f E C(X) such that f IK = g. Let {O,} be a collection of

open sets such that f1, is constant. Since U{O,} is dense, there is an a such that

X \ cl(D) n OQ = U 5 0. Then U is an open subset of K such that If(U)I = 1. This

is a contradiction. Therefore cl(D) = X and hence D is dense.
QED


Suppose now that X is a specker space and that 0 : X -- Y is a continuous

perfect irreducible map. Let g E C(Y). Then g o 0 E C(X). Since X is a specker

space, there is a K C X clopen and a 0 7 c E R such that g o 0(K) = c. But 0 is

irreducible so that int(O(K)) 5 0 is open. This proves the following lemma.










Lemma 3.5.5 If X is a speaker space and 0 : X -- Y is a continuous perfect irre-

ducible map, then Y is a weakly specker space.


For X, a compact Tychonoff space with a countable t7-base, we will construct a

quotient space that is an irreducible image of X and is also a metric space. This is

the content of the following theorem.


Theorem 3.5.5 If X is a compact Tychonoff space with a countable clopen it-base, then

there is a metric space Y and an irreducible surjection p : X -- Y. In particular,

EX- EY.


PROOF

Let II' = {B, : n E N} be a countable clopen 7r-base for X. Let I be the Boolean

algebra generated by I'. Then II is a countable clopen 7r-base for X. Define a relation

on X by x ~ y if x and y are in exactly the same sets of II. It is easily verified that

this is an equivalence relation on X. Let Y be the quotient space of X modulo this

equivalence relation and let p : X --+ Y be the projection map. Since p is continuous

and X is compact, Y is compact.

Let [x] denote the equivalence class of x in X. We claim that [x] = n{B E H : x e

B}. For if y E [x] then y E B for every B with x E B so that y E n{B E II : x e B}.

Conversely, if y n{B E II : x E B} then x E B implies y E B. Since II is a

Boolean algebra, if there is a B e 1 with y E B and x ( B, then X \ B E II has

y ( X \ B D n{B II : x e B}. This is a contradiction. Thus y e B implies x E B
so that y E [x]. In particular, since each B II is clopen any such intersection is

closed so that the points of Y are closed.

We will next show that the projection map is closed. By Proposition 1.6, Chapter

VI [10], it suffices to show that if K C X is closed, then p-'(p(K)) is closed. Let










K C X be closed, and let x E X \ p-l(p(K)). Then p(x) ) p(K), and since H is a
Boolean algebra, there is a B E II with x E B. Therefore, for every z E K there is a
B E II with x E Bz and z V Bz. Then {X \B, : z E K} is an open cover of K. Since
K is compact, there is a finite subcover say {X \Bi : 1 < i < n}. We then have that

x E nf=l Bi C X \ K and since each Bi is clopen, nl=1 Bi is clopen. It remains to
show that nf=l Bi C X \p-1(p(K)). Suppose that z E (n=l Bi) n p-1(p(K)). Then

p(z) = p(y) for some y E K. But then z E n7=l Bi implies y E nf=l Bi C X \ K;
a contradiction. Therefore nl=t Bi C X \p-'(p(K)) and X \ p-(p(K)) is open, so
that p-l(p(K)) is closed.
Now let K be a proper closed subset of X and suppose that p(K) = Y. Since
0 X \ K is open and II is a ir-base, there is a B E II with B C X \ K. Let x E B.
Since p(K) = Y, there is a y E K with p(y) = p(x), but this implies that y E B. A
contradiction. Therefore p is irreducible.
We now have that p is a continuous irreducible map from a compact Tychonoff
space X onto a compact T1 space Y. Then p is perfect and by Theorem 5.2, Chapter
XI [10], Y is regular.
We will next show that Y is metrizable by showing that Y is second countable and
appealing to a result due to P.Urysohn [10] which says for second countable spaces,
regularity is equivalent to metrizability.
Let y E Y. Then y = p(x) for some x E X. For every B E II with x E B, y =

p(x) E p(B) so that y E N{p(B) : B E n and x E B}. For B E H, B C p-'(p(B)). If
x E p-1(p(B)), then p(x) E p(B) so that p(x) = p(z) for some z E B. But then x E B
and therefore B = p-1(p(B)). Since B is open and the open sets U of Y are precisely
those for which p-1(U) is open, we have that p(B) is open for every B II Since
I is countable, {p(B) : B E H and x E B} is at most countable. Let y' E Y with

y' = y. Then y' = p(x') for some x' E X. Since y' # y, there is a B E I with x E B,










x' B. If p(x') E p(B) then there is a z E B with p(x') = p(z) and then x' E B.
Therefore, y' = p(x') i p(B) and as x E B, p(B) D N{p(B) : B E II and x E B}.
Therefore {y} = f{p(B) : B E H and x E B} so that every point of Y is a G6. In
particular, by Theorem 3.5(1)(1) [27], since Y is compact and every point is a G6, Y
is first countable.

Now let 0 0 U C Y be open and let y E U. By the above, there is a countable

subset of n say {Bi : i E N} such that {y} = n{p(Bi) : i E N}. We have already

seen that each p(B) is open for B E H and, since p is a closed map, each p(B) is
clopen. Then {Y \p(Bi) : i E N} is an open cover of Y \ U. Since Y \ U is closed and

hence compact, there is a finite subset, {Y \ p(Bi) : 1 < i < n} suitably reindexed,
with Y \U C U{Y \p(Bi) : 1 < i < n}. Then yE N{p(Bi) : 1 < i < n} C U

and n{p(B.) : 1 < i < n} is open. Let Bo = n{Bi : 1 < i < n}. Clearly Bo E Ii

and p(Bo) C n{p(B2) : 1 < i < n}. If z E n{p(Bi) : 1 < i < n}, then p-1(z) C

p-'({p(Bi) : 1 < i < n}) = {p-(p(Bi)) : 1 < i < n} = n{Bi: 1 < i < n} = Bo,
so that z E Bo. Therefore p(Bo) = ({p(Bi) : 1 < i < n}. This gives us that

{p(B) : B E II} is a countable basis for Y and therefore that Y is second countable.
We now have a perfect irreducible map p from X onto a metric space Y. Then

p o e : EX -+ Y is a perfect irreducible map from an extremally disconnected zero-
dimensional space onto Y. By the "uniqueness" of Theorem 3.4.2, EX 2 EY.

QED
As a corollary we have a partial converse of Theorem 3.5.1.

Theorem 3.5.6 Let X be a compact Tychonoff space with a countable 7r-base. If X

is a speaker space, then EX is a specker space. If this is the case, both X and EX

contain a countable dense set of isolated points.










PROOF

Let II be a countable 7r-base for X and let U E I. Since X is Tychonoff, there

is an f E C(X) with coz(f) C U. Since X is a specker space, there is a B clopen

and a 0 5 c R such that f(B) = c. In particular, B C coz(f) C U. If we choose

one such B for each U E I, then the resulting collection is a countable clopen r-

base for X. By Theorem 3.5.5, there is a metric space Y and an irreducible map

p : X --+ Y. By Lemma 3.5.5, Y is a weakly specker metric space. By Corollary 3.5.1,

Y contains a dense set of isolated points. Let D be this dense set of isolated points

and let t : EX -- Y be the irreducible map p o e as in the proof of Theorem 3.5.5.

By Lemma 3.4.1, t-'(D) is dense in EX. By Proposition 6.9(e) [27], the points of

1-I(D) are all isolated. Therefore, EX contains a dense set of isolated points, and

so EX is a specker space. Let I(EX) and I(X) denote the set of isolated points of

EX and X respectively. Again, by Proposition 6.9(e) [27], eII(EX) is a bijection from

I(EX) onto I(X). If H is a ir-base for X, then II must contain I(X). Therefore if H

is countable, then I(X) and hence I(EX) are countable.
QED


Recall that a space is locally compact if each point has a compact neighborhood.

In Chapter 6 [15], it is shown that X is open in #X if and only if X is locally compact.

This gives most of the following corollary.


Corollary 3.5.2 Let X be a locally compact Tychonoff space with a countable r-base.

If X is a specker space, then EX is a specker space.


PROOF

Let II be a countable r-base for X. Let 0 $ 0 C iX be open. Since X is locally

compact, by the above observation, and since X is dense in #X, On X is a nonempty

open subset of X. Then there is a B E H with B C O n X. Again, as X is open in











3X, B is open in 3X so that II is a countable ir-base for fX. Then fX is a compact

specker space with a countable r-base so that by Theorem 3.5.6, E(/3X) is a specker

space. Since E(/X) -/3(EX), EX is a specker space.
QED


To return to the standing of Conjecture 3.5.1, we begin with the following result

which will allow us to restrict our attention to compact zero-dimensional spaces.


Lemma 3.5.6 If X is a compact quasi-specker space then there is a compact zero-

dimensional space Y such that EX EY.


PROOF

Since X is a quasi-specker space, S(X) is o-large in C(X). By Theorem 3.4.3,

the map 0: Max(C(X)) -- Max(S(X)) isdn irreducible. Since X is compact, X -

Max(C(X)) and since S(X) is hyper-archimedean, Max(S(X)) is zero-dimensional.

QED

Now if X has the countable chain condition and 0 : X -- Y is any continuous map,

then Y has then countable chain condition. Therefore if X is a compact quasi-specker

space with the countable chain condition, then X is co-absolute with a compact zero-

dimensional space Y with the countable chain condition. The advantage of dealing

with compact zero-dimensional spaces is that in the context of the following theorem,

due to A. Hager, we can translate our topological problem to one of Boolean algebras.


Definition 3.5.3 Let A be a Boolean algebra. For C, D C A we say that C refines D,

denoted by C -< D, if for every c C there is a d E D such that c < d.


Definition 3.5.4 Let A be a Boolean algebra. A quasi-cover of A is 0 C C A with

VC = 1. A cover is a finite quasi-cover. A partition is a cover by pairwise disjoint

elements.










Definition 3.5.5 A Boolean algebra is called speaker if every sequence of covers has a

common refinement by a quasi-cover.

We will need the following lemma.

Lemma 3.5.7 A zero-dimensional space X is a specker space if and only if for every

0 : f E C(X) there is a C C B(X) such that UC is dense and for every C E C, fic
is constant.


PROOF
Suppose that X is a specker space and let 0 : f E C(X). Then there is a clopen

K C X such that f K is a nonzero constant. Let C = {K E B(X) : f K is constant).

If UC is dense, we are done. Suppose that UC is not dense. Then int(X \ (UC)) 7 0

is open. Since X is zero-dimensional, there is a 0 # T clopen with T C int(X\ (UC)).

Let g = fXT. Then 0 # g E C(X), and g is not a non-zero constant on any clopen

subset. This is a contradiction. Therefore UC is dense. The converse is clear.
QED


It should be pointed out that for the above result it is sufficient that X have a

clopen 7r-base. With this observation the preceding lemma is actually a restatement

of Proposition 0.5 [23]; X is a specker space if and only if it is a weakly specker space

with a clopen r-base.

Suppose that {Ci : i E N} is a sequence of covers of A. Suppose also that for

each i E N, Ci = {cj : 1 < j < ni}. Let di = cl and, for 1 < j < ni, let

dj = cj \ V{ck : 1 < k < j 1}. Let 2Di = {dj : 1 < j < ni}. Then Di is a partition

of A and AI -< Ci. If D is a common refinement of the sequence {Di : i E N} by a

quasi-cover, then Z> is a common refinement of the sequence {Ci : i E N} by a quasi-

cover. Define 'D inductively by VDo = Do and VYn+ = {a A b: a E VYn and b E ++ }.










Then {12 : i E N} is a sequence of partitions with D -< Di such that D' >- D' > .

If D is a common refinement by a quasi-cover of the sequence {DfI : i E N}, then D is

a common refinement by a quasi-cover of the sequence {Di : i E N}. Call a sequence

of covers {D, : i E N} with D1 >- D2 >- a decreasing sequence of covers. We have

proved the following lemma.

Lemma 3.5.8 A Boolean algebra A is specker if and only if every decreasing sequence

of partitions has a common refinement by a quasi-cover.

We are now ready to state and sketch the proof of the following theorem which

is due to A. Hager.

Theorem 3.5.7 A Boolean algebra A is specker if and only if its Stone dual St(A) is

a specker space.

PROOF

Suppose A is is specker. Let 0 7 f E C(St(A)). Since St(A) is compact and

zero-dimensional, there is a finite cover of St(A) by clopen sets such that f varies

less than 2-" on each set of this cover. For each n, let U4 be this cover. Then in

B(St(A)) A, {1U : n E N} is a sequence of covers. By the hypothesis, there is a

common refinement by a quasi-cover U. Then VU = cl(UU) = St(A) so that UU is

dense. Since U -< U4 for all n, if K E U then f varies less than 2-" on K for all n.

Therefore is f is constant on K.

Suppose now that St(A) is a specker space. Let {Ci} be a sequence of covers

of A. By Lemma 3.5.7, we may assume without loss of generality that {Ci} is a

decreasing sequence of partitions. Via the isomorphism A B(St(A)) we may view

these as a sequence of finite clopen partitions of St(A). We then construct a Cantor

type function f E C(St(A)) as the uniform limit of simple functions f, defined on C1










having the property that if Ci E Ci has C1 D C2 D .. then there exists an r E R

such that nCi = f-l(r) and if r E f-(St(A)) then f-'(r) is of this form. By the

hypothesis and Lemma 3.5.6, there exists a C C St(A) with UC dense such that

for every C E C, f\C is constant. Viewed in A, C is a quasi-cover and a common

refinement of the sequence {Ci}.
QED


For our purposes, we have that a compact zero-dimensional space X is a specker

space if and only if B(X) is a specker boolean algebra. This is where we can make

a translation of our original problem. All of the results used in this paragraph can

be found in [27] Chapters 3 and 6. Suppose that X is a compact zero-dimensional

specker space with the countable chain condition. Then B(X) is a specker boolean

algebra with the countable chain condition. The irreducible map e : EX -- X

induces an o-large embedding of B(X) in B(EX), and a boolean isomorphism of R(X)

and R(EX). Since EX is extremally disconnected, R(EX) = B(EX). Identifying

isomorphic algebras, we have an o-large embedding of B(X) in 7R(X). Since R(X)

is the completion of B(X), our original question translates to the following. Is the

completion of a specker boolean algebra with the countable chain condition a specker

boolean algebra? What follows is part of an attempt to answer this question.

Definition 3.5.6 Let {C : i E N} with C1 >- C2 >- be a decreasing sequence of

covers of A. C1 >- C2 >- -.. is said to be a binary sequence of covers if for every i E N

and for every c E CC, I{b C,+i : b < c}1 < 2.


We will show that specker boolean algebras can be characterized in terms of binary

sequences of covers. We first need some terminology and notation. For a boolean

algebra A, C C A is called a chain if C is a linearly ordered subset of A. For a

decreasing sequence of covers C1 >- C2 >- *, a chain cl > c2 > is a representative










chain if c, E Ci for all i E N. A chain cl 2 c2 > .. is called a proper chain if there

exists an i = j such that c; = cj, provided that {Ci} is not a trivial sequence. For

a e A, let [0,a] = {b E A : 0 < b < a}. For r E R, let [rJ denote the largest integer

less than or equal to r and let Fr] denote the least integer greater than or equal to r.

We first need the following technical lemma.


Lemma 3.5.9 Let A be a boolean algebra. If for every binary sequence of covers

B1 >- B2 > there is a proper representative chain { bi} with a nonzero lower bound,
then for every sequence of covers C1 >- C2 >- ... there is a proper representative chain

{ci} with a nonzero lower bound

PROOF

By way of contradiction, suppose that Ci >- C2 > ... is a sequence of covers such

that every proper representative chain {ci} has A{ci : i E N} = 0. We will construct

a binary sequence of covers with the same property.
The idea of this proof is to construct, for every C >- Ci+1, a binary sequence of

covers between" Ci and Ci+l. We do this pointwise for each element a of Ci by

taking pairwise joins of the elements of Ci+1 that are below a.

Let i E N and for a E Ci, let Ci+i(a) = {c E Ci+1 : c < a}. Then Ci+i(a) is

a binary sequence of covers of [0,a]. Suppose that we have constructed Bl(a) >-

B'_1 (a) > >.- 3z(a) = Ci+l(a) a binary sequence of covers of [0, a]. Construct B'

as follows. First there is an indexing of the elements of Bk(a) = {b, : 1 < n < mk}.

For 1 < n < ["Lj, let an = b2n-1 V b2n and if mk is odd, let arJ-i = b,,. Now take

S= {a 1 < n < [ 1 }. For some finite k E N, 8 = {a}. Let B = B'n for
1 < n < k. We have constructed {a} = Bi(a) >- B2(a) >- >- Bk(a) = Ci+l(a) a

binary sequence of covers of [0, a]. Now repeat this process for each a E Ci. Suppose

that Ci = {an : 1 < m < mi}. Let kn be such that Bkn(an) = Ci+i(an) for 1 < m <










mi. Let k = max{k,n : 1 < n < mi}. We need to do some manipulation of the indices.

For each 1 < n < mi and for each j E N with k, < j < k, let Bj(a,) = Bkn(a,).

Now for 1 < 1 < k, let BI = Bi(al) U B2(al) U ** U Bi(am,). We have constructed

Ci = B1 >- B2 >- *. >- k = Ci+l a binary sequence of covers of A. Repeat this
process for every pair C, >- Ci+1 of the original sequence of covers and reindex the

resulting binary sequence so that C1 = B3 >- B2 >-- .. Then for each Ci there is an ij

such that Ci = Bi,. Now let {a,} be a proper representative chain of the sequence B,.

By construction there is a chain {aj } with {ai,} E C, which is cofinal in {as}. Since

{a,} is a proper chain and {ai, } is cofinal, {ai, } is a proper representative chain of
the sequence {Ci}. By the hypothesis, A{a,,} = 0. As any lower bound for {an} is a

lower bound for {a, }, we have that A{a,} = 0
QED


We then have the following theorem which allows us to characterize specker

boolean algebras in terms of binary sequences of covers.

Theorem 3.5.8 Let A be a boolean algebra. A is specker if and only if for every binary

sequence of partitions B3 >- B2 >- '* there is a proper representative chain {bi} such

that {b,} has a nonzero lower bound.


PROOF

Suppose that A is specker. Let B1 >- B2 >- be a binary sequence of partitions.

Let B be a common refinement by a quasi-cover. Since VB = 1, there is a 0 / b E B.

Since B is a common refinement, for each Bi there is a bi E Bi with b < b,. Since each

Bi is a partition, {bi} is a representative chain. If B1 >- 2 >- has no representative

chains {b,} with bi / bj for some i 5 j, then we can take B = B1 and the sequence is

trivial.










Suppose that C1 >- C2 >- is a decreasing sequence of partitions of A. By

Lemma 3.5.6, it suffices to show that this sequence has a common refinement by a

quasi-cover. By the hypothesis and Lemma 3.5.7, there is a proper representative

chain {c~ : i E N} such that {c, : i E N} has a nonzero lower bound. Let L be the set

of all nonzero lower bounds for all proper representative chains of C1 >- C2 >- -... Then

L is a common refinement of the Ci's. It remains to show that L is a quasi-cover. If

VL = 1 we are done. Suppose that VL # 1. Then there is an a E A with 0 < a < 1

such that a is an upper bound for L. Let b = a' and let C,[b] = {cA b: b E C;}. Then

Ci[b] is a decreasing sequence of partitions of [0, b]. Now, Cl[b]U {a} > C2[b]U {a} >- **

is a decreasing sequence of partitions of A. Therefore, by the hypothesis and Lemma

3.5.9, there is a proper representative chain {bi} with a nonzero lower bound. Since

a A b = 0, bi E Ci[b] for all i. For each i, bi = ci A b for some ci E Ci. Since {bi} is

a chain and each Ci is a partition, {ci} is a chain. Since {bi} is a proper chain, so

is {c;}. If d is a nonzero lower bound for {bi} then it is also a lower bound for {c;}

and therefore d < a. This is a contradiction since d < b and a A b = 0. Therefore,

V L = 1 and the proof is complete.
QED


The current goal with these results is to show that if X is a compact specker space

with the countable chain condition, then X contains a dense set of isolated points.

We begin by assuming that X has no isolated points. Then B(X) is an atomless

specker boolean algebra. It may be possible to then construct an uncountable anti-

chain, violating the countable chain condition. This will show that X contains one

isolated point. It should then be possible to argue that X in fact contains a dense

set of isolated points, and therefore that EX is a specker space.















CHAPTER 4
CONCLUSION

The focus of this dissertation is an examination of f-rings which are rich in idem-

potents. We consider three different types of f-rings, reflecting three different degrees

of "richness".

In Chapter 2, we consider local-global f-rings. For bounded rings, we obtain the

equivalence of the following conditions.

1. A is a local-global ring.

2. Every primitive quadratic polynomial with non-negative coefficients in A rep-

resents a multiplicative unit.

3. Max(A), the maximal spectrum, is zero-dimensional.

As one measure of the richness of idempotents for local-global f-rings, recall that

in the verification of the local-global condition, we began with a primitive polyno-

mial f E A[t] and constructed an element s E A such that f(s) is a multiplicative

unit. This element s we constructed is, in fact, a linear combination of idempotents.

The zero-dimensionality of Max(A) also gives another indication of the richness of

idempotents in local-global rings. Here we need to recall that the clopen subsets of

Max(A) are the basic open sets determined by an idempotent of A and that the

zero-dimensionality of Max(A) implies that the clopen sets are a base for the open

sets of Max(A). Considering the case A = C(X) for a Tychonoff space X, we obtain

a nice addition to the "Algebra-Topology Dictionary". Namely, that a space X is

strongly zero-dimensional if and only if every primitive quadratic polynomial with
87










non-negative coefficients in C(X) represents a unit. This characterization is of addi-

tional interest because it provides a first order algebraic characterization of strongly

zero-dimensional spaces. For this reason, this condition merits further investigation.

Chapter 2 also left a glaring open problem. That is, whether or not all of the above

conditions are equivalent in the unbounded case. As indicated at the end of the

chapter, the problem is one of "cutting down" primitive polynomials to the bounded

subring A(1) of A and maintaining primitivity. We give sufficient conditions for this

to occur, namely, that the ring be a Bezout ring. The working conjecture is that

the general result does hold, but the proof may require some sort of set theoretical

forcing argument.

In Chapter 3, we consider two related measures of the richness of idempotents.

Recall that an f-ring A is said to be of specker-type if A is an essential extension

of S(A), the subalgebra generated by the idempotents of A, and A is said to be a

quasi-specker ring if A is an o-essential extension of S(A). Recall also that for a ring

A, Q(A) denotes the complete ring of quotients of A. We first obtain the equivalence
of the following conditions.

1. A is a specker-type ring.

2. A is an f-subring of Q(S(A)).

3. S(A)L = Q(A).

In particular, condition (3) gives some indication of the richness of idempotents in

specker-type f-rings. It says, in effect, that every element of the complete ring of

quotients of A can be written as a (possibly infinite) linear combination of idem-

potents. For quasi-specker f-rings, the richness of idempotents is indicated by the

defining condition that every nonzero element is larger than some scalar multiple of a










nonzero idempotent. Results regarding specker-type and quasi-specker rings become

particularly interesting for the case A = C(X). Recall that a space X is said to

be a specker space (quasi-specker space) if C(X) is a specker-type (quasi-specker)

ring. For specker spaces, we have in addition to the previous conditions, suitably

translated, the following.

4. For every nonzero f E C(X), there is a clopen set on which f is nonzero and

constant.

In the case that X is compact we are able to improve this characterization to,

5. For every f E C(X) there is a collection of clopen sets whose union is dense in

X and such that f is constant on each of these clopen sets.

Again, confusing the clopen sets of X with the idempotents of C(X), we get some

sense of the richness of idempotents for specker spaces. For compact quasi-specker

spaces, the richness of idempotents of C(X) is reflected in the condition which char-

acterizes these spaces as those which have a countable clopen 7r-base.

Motivated by the "tight" containment of C(X) in Q(X), we consider the specker

condition for spaces vis-a-vis their absolutes. We are able to show that if a space X

is a quasi-specker space and EX, the absolute of X, is a specker space, then X is

a specker space. As a partial converse, we have the conjecture that if X is compact

with the countable chain condition and X is a specker, then EX is a specker space.

As a special case of this conjecture we are able to show that if X compact, with

a countable r-base, then X is a specker space implies that EX is a specker space.

Finally, via A. Hager's result on specker boolean algebras and specker spaces, we are

able to translate the problem of compact specker spaces to one of specker boolean

algebras. Here, using the characterization obtained for specker algebras in terms of







90


refinements of binary sequences of covers, we hope to prove the following. If X is a

compact specker space with the countable chain condition, then X contains a dense

set of isolated points. Then EX will also contain a dense set of isolated points and

therefore be a specker space.















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BIOGRAPHICAL SKETCH

Scott David Woodward was born in Phoenix, Arizona, in 1955. Before returning

to school in 1980, he was a journeyman mason working out of Orlando, Florida. He

received his B.S. and M.S. degrees in mathematics from the University of Florida in

1983 and 1987 respectively. He is married to Rita Wendt-Woodward and has two

children, Christopher and Michael.










I certify that I have read this study and that in my opinion it conforms to accept-
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.





Jorge Martinez, Chairman
Professor of Mathematics




I certify that I have read this study and that in my opinion it conforms to accept-
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.





Krishnaswami Alladi
Professor of Mathematics




I certify that I have read this study and that in my opinion it conforms to accept-
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.


Douglas 4. Center
Professor of Mathematics


---


uJi~'~~









I certify that I have read this study and that in my opinion it conforms to accept-
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.






Christopher W. Stark
Associate Professor of Mathematics




I certify that I have read this study and that in my opinion it conforms to accept-
able standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.






El by J. olduC Jr.
Professor of Instruction and Curri ulum




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.




August 1992


Dean, Graduate School