Generalizations of ideal theory

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Generalizations of ideal theory
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Ideals (Algebra)   ( lcsh )
Rings (Algebra)   ( lcsh )
Lattice ordered groups   ( lcsh )
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Mathematics thesis Ph. D
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Thesis (Ph. D.)--University of Florida, 1982.
Bibliography:
Bibliography: leaf 72.
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Typescript.
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Vita.
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by David B. Kenoyer.

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Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
    Table of Contents
        Page iii
    Abstract
        Page iv
        Page v
    Chapter 1. Preliminaries
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
    Chapter 2. Commutative multiplicative ideal lattices
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
    Chapter 3. Multiplicative subsets, lattices of extensions, and localizations
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
    Chapter 4. Invertible elements
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
    Chapter 5. Generalizations of valuation rings and prufer domains
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
    Chapter 6. Generalization of dedekind domains
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
    Chapter 7. Modularity
        Page 70
        Page 71
    Bibliography
        Page 72
    Biographical sketch
        Page 73
        Page 74
        Page 75
Full Text















GENERALIZATIONS OF IDEAL THEORY


BY


DAVID B. KENOYER






























A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


1982















ACKNOWLEDGEMENTS

The author would like to thank his Supervisory

Committee Chairman Jorge Martinez in particular, and all

of the members of his supervisory committee for their

helpful comments, criticisms, and suggestions. He also

would like to acknowledge the cooperation of his typist,

Sharon Bullivant, who did such quality work in such a short

time.















TABLE OF CONTENTS

Page
ACKNOWLEDGEMENTS ...................................... ii

ABSTRACT .............................................. iv

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

CHAPTER

I. PRELIMINARIES .................................. 4

II. COMMUTATIVE MULTIPLICATIVE IDEAL LATTICES ...... 1

III. MULTIPLICATIVE SUBSETS, LATTICES OF EXTENSIONS,
AND LOCALIZATIONS .............................. 22

IV. INVERTIBLE ELEMENTS ............................ 39

V. GENERALIZATIONS OF VALUATION RINGS AND PRUFER
DOMAINS ........................................ 48

VI. GENERALIZATION OF DEDEKIND DOMAINS ............. 60

VII. MODULARITY..................................... 70

BIBLIOGRAPHY ............................................ 72

BIOGRAPHICAL SKETCH ................................... 73


iii













Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

GENERALIZATIONS OF IDEAL THEORY

By

David B. Kenoyer

August 1982

Chairman: Dr. Jorge Martinez
Major Department: Mathematics

The lattice of ideals of a commutative ring and the

lattice of convex -subgroups of a lattice-ordered group

fit into the more general setting of a mi-lattice, which is

an algebraic lattice with an associative multiplication that

distributes over joins, respects compactness, and satisfies

the ideal multiplication property that ab s a A b. In this

context, the terms primary element, residual quotient,

multiplicative element, cancellative element, and invertible

element are defined, and generalizations of classical

results of commutative ring theory are obtained. A primary

decomposition theorem for modular mi-lattices with ACC and

enough multiplicative elements is given. Multiplicative

subsets, rings of quotients, and extensions and contractions

of ideals are generalized, as are commutative rings with

identity, integral domains, and valuation rings. Several








characterizations of the analogue of PrUfer domains are

given in terms of localizations and arithmetic relation-

ships. Dedekind domains are generalized and characterized.














INTRODUCTION

This dissertation is an attempt to place the results

of multiplicative ideal theory in a lattice-theoretic

setting which will include .other lattices of subobjects

as well. The two motivating examples we will keep in mind

are the lattice of ideals of a commutative ring and, to a

lesser extent, the lattice of convex -subgroups of a

lattice-ordered group.

We consider an algebraic lattice L endowed with a

multiplication which distributes over all joins, respects

compactness, and satisfies the ideal multiplication property

that ab a A b. A major work in this area is that of

Keimel [5]. He indicates that this approach applies to a

wide class of known examples, including the lattices of

ideals of commutative semigroups, distributive lattices,

and f-rings. He has developed the theory of minimal primes

to a great extent, and so in Chapter I, after we introduce

the basic concepts, we list his major results, and then turn

to other classical results of commutative ring theory. Our

terminology differs slightly from that of Keimel, and

beginning in Chapter II, we assume that our multiplication

is associative and commutative, as it is in our two models.

Modularity is only assumed for one result, and we do not

know that it is necessary there.

1





2

Chapter II considers primary elements and primary

decompositions, ending with Theorem 2.14, the analogue of

the Primary Decomposition Theorem for commutative Noetherian

rings.

Chapter III is an investigation of multiplicative sets

and our version of rings of quotients. We consider the

"extension" a of an element a in our lattice relative to a

multiplicative subset M of compact elements, and show that,

in passing from the lattice L to the lattice of extensions,

LM, we preserve compactness, primes, primary elements,

radicals, and almost all finite arithmetic operations,

including all operations with compact elements. Theorem

3.22 establishes the fundamental relationship between an

element a and the extensions of a relative to the maximals

exceeding a.

In Chapter IV, we discuss invertibility, including

various arithmetic facts involving invertible elements and

their existence in L. We conclude Chapter IV with a dis-

cussion of various interpretations of the concept of an

integral domain. Chapter V looks at results from the theory

of Prefer domains, including the characterizations in terms

of arithmetic relations, which comprise Theorem 5.10. We

also consider our version of a valuation ring, and its

relation to Priifer domains. Chapter VI deals with our

generalization of Dedekind domains, with the main result

being Theorem 6.3. Finally, Chapter VII briefly looks at

the question of modularity, giving a result which leads to

certain conditions that guarantee modularity.








As general references for the multiplicative ideal

theory of commutative rings, we suggest Gilmer [3] or

Hungerford [4]. For the theory of lattices, we refer to

Birkhoff [2], while for the theory of lattice-ordered

groups, we use Bigard et al. [11, which is written in

French.

Concerning notation, we will use Z to stand for the

set of natural numbers, and R to stand for the set of real

numbers.














CHAPTER I
PRELIMINARIES

Let L be a complete lattice with minimum element 0 and

maximum element 1 ; 0. An element c of L is said to be
n
compact if whenever c < v a. it must be that c < v a. .
iEI k=l 1k
0 is automatically compact, and the join of any finite set

of compact elements is again compact.

A lattice L is algebraic if it is complete and every

element of L is the join of a (possibly infinite) set of

compact elements. In an algebraic lattice, an element c is

compact if and only if whenever c = v a. it must be that
n iEIl
c v a.
k=l k


1.1. Definition: A lattice L is a multiplicative

ideal lattice, or mi-lattice, if L is an algebraic lattice

endowed with an associative binary operation (a,b) ab

that satisfies the following conditions:

(1) ab s a A b for all a,b E L.

(2) a( v bi) = v (ab.) and ( v bi)a = v (bia) for
iEI iEI -- i ieI

all a E L and for all families {bili E I} in L.

(3) If x and y are compact in L, then xy is compact

in L.








1.2. Example: Let L be the lattice of ideals of a

ring R. L is a mi-lattice if we take the meet to be inter-

section, the join to be the sum, and multiplication to be

the ordinary product of ideals. Here, the compact elements

are the finitely generated ideals.



1.3. Example: A lattice-ordered group is a group G

endowed with a lattice order which is compatible with the

group operation in the sense that a(b v c) = ab v ac,

(b v c)a = ba v ca, and the dual conditions involving meets

also hold. An k-subgroup of G is a subgroup H which is

also a sublattice; i.e., if a and b are in H, then a v b and

a A b are also in H. An -subgroup C of G is convex if

whenever a and b are in C and x E G satisfies a x b,

then x E C. A theorem of Bigard et al. [i], states that

the lattice C(G) of convex Z-subgroups is completely dis-

tributive, if we let the join of two convex Z-subgroups be

the convex k-subgroup generated by their union. Thus, if

we let the product and meet coincide, C(G) is a mi-lattice.

Again, the compact elements are the finitely generated ones,

and in this case, these are generated by a single element.



1.4. Definition: A lattice is said to have the

ascending chain condition, or ACC, if every ascending chain

breaks off after a finite number of steps; i.e., if

x 1 x2 : x3 : ..., then there is a positive integer n with

xk = xn for all k n. A result from lattice theory








(Birkhoff [21) is that the following conditions are equiva-

lent for an algebraic lattice L:

(a) L has ACC.

(b) Each element of L is compact.

(c) Each nonempty subset of L has a maximal element.

In both Examples 1.2 and 1.3, there is a notion of a

prime element. In Example 1.2, recall that a prime ideal

is a proper ideal P of a ring R satisfying the condition that

if A and B are ideals and AB c P, then A c P or B c P. In

Example 1.3, a prime convex -subgroup is a proper convex

i-subgroup P such that if A and B are convex k-subgroups of

G and if A n B S P, then A c P or B c P. These two notions

of primes motivate the following definition.



1.5. Definition: If L is a mi-lattice and p E L, then

p is prime if p < 1 and whenever ab p, we must have a p

or b p. Notice that if we want to show that an element is

prime, it is enough to show that if a and b are compact and

ab p, then a 5 p or b p. Conversely, if p is prime,

then certainly the definition of a prime applies when a and

b are compact elements, so that we may assume a and b are

compact in the definition.

An element m of a lattice L with 1 is a maximal if it

is maximal in the set of elements strictly below 1; i.e., if

m a < 1, then m = a. Recall that in a commutative ring

with identity, every proper ideal is contained in a maximal

ideal, and each maximal ideal is prime. In our context, we

have the next three results.








1.6. Proposition: In a lattice L with 1, if 1 is com-

pact, then each element a < 1 lies below a maximal of L.



Proof: Let a < 1, and let A = {bfa b < 1}. A ,

since a E A. A is partially ordered as a subset of L. If

C = {cili E I} is a nonempty totally ordered subset of A,

then let c = v c. We have a c, so suppose c 4 A. Then
iEI
n
c = 1, so 1 = v c.. Since 1 is compact, 1 = v c. with
iEI i k=l 'k
ik E I for 1 k s n. But C is totally ordered, so

1 = c0 E C, which is impossible. Thus c E A, so by Zorn's

Lemma, there is an element m which is maximal in A. But if

m : b < 1, then b a, so b e A and b = m. Hence m is a

maximal of L.D



1.7. Example: The converse to Proposition 1.6 is, in

general, false. The following example and ideas are found

in Bigard et al. [1]. Let X be a locally compact space

which is not compact, and let L(X) = {f: X Rjf is locally

constant with compact support}. Then the only prime convex

Z-subgroups of L(X) are those of the form C = {f E GIf(x) =0}

for some x E X. Each of these is maximal, and each convex

Z-subgroup of L(X) is contained in one of these. Also, each

f E L(X) must be in Cx for some x E X, since X is not compact,

and so L(X) cannot be generated by a single element; i.e.,

in C(L(X)), 1 is not compact. A lattice-ordered group G in

which all primes are maximal is called hyper-archimedean,

and an element a e which generates a lattice-ordered group






8

G is called a strong order unit of G. Any hyper-archimedean

lattice-ordered group without a strong order unit will be

a counterexample.


2
1.8. Proposition: In a mi-lattice L, if 1 = 1, then

each maximal is prime. If each a < 1 lies below a maximal,
2
and if each maximal is prime, then 1 = 1.


2
Proof: If 1 = 1, let m be a maximal of L. Since

m < 1, we suppose x and y are in L with xy m, but x m.

Then l(m v y) = (m v x)(m v y) = m2 v xm v my v xy 5 m < 1,

so m v y < 1. But m v y m, so m v y = m and y < m. Thus

m is prime. On the other hand, if each a < 1 lies below a

maximal, and if each maximal is prime, then 12 cannot lie
2
below any maximal, so 1 = l.D



1.9. Definition: The radical of an element a, denoted

rad(a), is defined as rad(a) = A{plp is prime and a 5 p}.

By convention, we set Ap = 1 and v p = 0. A result of
PE- petp
Keimel [5] is:



1.10. Lemma: If M is a nonempty set of nonzero compact

elements of a mi-lattice L, and if M is closed under multi-

plication, and if a E L with m $ a for each m E M, then

Sa = {blb > a and m $ b for each m E M} must have maximal

elements, and each of these is prime in L.








1.11. Proposition: If L is a mi-lattice and a E L,

then rad(a) = v{xfx is compact and xn < a for some positive

integer n}.



Proof: We show that {xlx is compact and x s rad(a)} =

{xjx is compact and for some positive integer n, xn 5 a}.

First, if x is compact and xn s a, then x S p for each prime

p a, so {x compactlxn S a for some n} S {x compactjx

rad(a)}. But if x is compact and if, for each n E Z ,
X n$ a, then {xnin 11} is a multiplicatively closed set of

nonzero compact elements, and so if Sa = {b E Lib a .and
na
x n b for all n E Z }, then S has a maximal element p which
a
is prime, by Lemma 1.10. Hence p0 a and x t p0, so

{x compactjx s rad(a)} = {x compact for some n > 1, xn s a}.D



Keimel [5] defines a m-semiprime element of a mi-lattice

L to be an element x such that for all t E L, t $ x implies
2
t 2 x. He shows that each m-semiprime element in a mi-

lattice is the meet of the set of prime elements exceeding

it (Theorem A), and so these are the elements that occur as

radicals in mi-lattices. As a corollary, he shows in a

mi-lattice L that the following conditions are equivalent:

(1) Each element of L is the meet of primes, so that

rad(a) = a for all a E L.

(2) Each a E L is semiprime.
2
(3) For each a E L, a = a.

(4) For all a,b E L, ab = a A b and L is distributive.






10

(5) For all a,b E L, ab = a A b and L is completely

distributive.

(6) For all a,b E L, ab = a A b and L is Brouwerian.

In ring theory, these conditions on ideals of a commutative

ring R are equivalent to R being von Neumann regular, by

a result of Kaplansky. Keimel then looks at semiprime mi-

lattices, which are mi-lattices in which 0 is semiprime.

He defines the pseudo-complement of an element, and m-pre-

filters and m-filters, and uses these and the Gratzer-Schmidt

isomorphisms between elements of L and ideals of the set of

compact elements of L to set up a bijection between maximal

m-filters and minimal primes of a mi-lattice L which is

semiprime and satisfies the property that for all r, s, and

t E L, r A s = r A t = 0 implies (r v s) A t = 0. This gives

him several characterizations of a semiprime pseudo-comple-

mented lattice. He goes on to look at the set of minimal

primes and develops the hull-kernel topology, showing that

it gives a completely regular space, with a basis of closed-

and-open sets determined by the compact elements of L. He

finishes with a discussion of z-elements, which area gener-

alization of z-subgroups of a lattice-ordered group, and uses

them to describe certain classes of mi-lattices, including

those in which the space of minimal primes is compact. Keimel

actually does all this in a more general setting, frequently

not requiring that the product of compact elements be compact,

or that the product distribute over joins, or that it be

associative.














CHAPTER II
COMMUTATIVE MULTIPLICATIVE IDEAL LATTICES

For the remainder of this dissertation, L will always

represent a mi-lattice with a commutative multiplication.

We will consider the analogues of some classical results of

multiplicative ideal theory.



2.1. Definition: The residual quotient of a by b, de-

noted by [a:b], is defined by: [a:b] = v{clbc < a}. Note

that if x is compact and x [a:b], then bx a, so

[a:b] = v{x compactlbx a}.



2.2. Proposition: The following properties of residual

quotients hold in L:

(a) [a:b] z a, [ab:b] a, and b[a:b] < a A b, for all

a,b E L.

(b) If b a, then [a:b] = 1.

(c) [[a:b]:c] = [a:bc] for all a,b,c e L.

(d) [a:b] = [a:a v b] = [a A b:b], for all a,b E L.

(e) [a: v b.i] = A [a:b.], for all a E L and all families
iEI ieI 1
(bi)E L.

(f) [( A ai):b] = A [ai:b], for all b E L and all
iEI iEI
families (a.). a L.








(g) [ab:c] a[b:c], for all a,b,c E L.

(h) [a: A b ] v [a:b.] for all a E L and families
ieI iE I
(bi)i eI c L.

(i) [( v a.):b] v [a.i:b] for all b E L and families
iEI iEI
ie l 1 e L.
(a) el C L.


Proof: (a) ab < a, so a < [a:b]; ab < ab, so

a -< [ab:b]; b[a:b] = b(v{clbc < a}) = v{bclbc < a} < a, and

b[a:b] < b.

(b) First b-l < b, so if b < a, b-1 :- a, and [a:b] = 1.

(c) [[a:b]:cl = v{dlcd < [a:b]} = v{dlbcd < a} = [a:bc].

(d) We have (a v b)c < a 4> ac v bc a 4 bc < a <=>

bc a A b.

(e) [a: v b.]( v bi) s a, so for each j E I,
iI ilEI
[a: v b.]b. a, and [a: v b. i < A [a:b.]. But
il J el 1 il
i iEI i iEI

A [a:bi ]) ( v b.) = v (( A [a:b ])bj) < v ([a:bib. ) a,
iEl 1 jEI jEl iel jEl I

and so A [a:b.] = [a: v bi.].
iel iel 1

(f) First, b[( A a) :b3 < A a. :- a. for each j E I,
iEI iEl 1 3
so [( A a.):b] < A [a.:b]. But b( A [a.:b]) b[a.:b] < a.
iel =el iel 1- J

for each j E I, so [( A a.):b] = A [a.:b].
iel 1 ie I

(g) This is true since a[b:c] = a(v{dlcd < b}) =

v{adlcd -< b} < v{adlcad s ab} [ab:c].








(h) ( A b ) ( v [a:b.]) = v (( A b.)[a:b.])
iel jEI jeI iEI

v (b.[a:b.i1) a.
jEI J J

(i) Since b( v [a.i:bl) = v (b[a.i:b]) v a., we have
iEI iEI iEI
1e el xilET

v [a :b] : [ai:b] < [( v a.):b].D
iel11il 1
EI iEI



2.3. Definition: An element a E L is multiplicative if

a[b:a] = b for each b < a; i.e., if a divides every element

below it. Note that if a is multiplicative, then a-l = a.

An element a E L is said to be power-multiplicative
n +
if a is multiplicative for each n E Z

Considering our two examples of mi-lattices, if R is a

commutative ring, all principal ideals are multiplicative,

and in fact power-multiplicative. If G is a lattice-ordered

group, then since multiplication is the meet operation,

every convex k-subgroup is multiplicative, and hence power-

multiplicative. Thus, in the two underlying models, each

element of the lattice is the join of a set of compact,

power-multiplicative elements. For this reason, we will

assume this property to obtain one nice result later in

this section, Theorem 2.14, without losing touch with our

motivations.

The first question that arises is: When is the product

of two multiplicative elements again a multiplicative element?

The answer does not seem to be that it is always true, but a

partial answer is offered.








2.4. Lemma: If a and b are multiplicative and if

b = [ab:a], then ab is multiplicative.



Proof: Let c ab. Then c < a, so a[c:a] = c. Thus

a[c:a] ab, so since b = [ab:a], [c:a] < b. Hence

[c:a] = bd, where d = [[c:al:b] = [c:ab], and

ab[c:ab] = abd = a[c:a] = c.0



We now press on toward a primary decomposition result.

Recall that in a commutative ring R, an ideal Q c R is

primary if whenever xy E Q with x and y elements of R, then

x E Q or yn E Q for some positive integer n. Since a compact

ideal is finitely generated, we offer the following.



2.5. Definition: An element q in a mi-lattice is primary

if q < 1 and whenever x and y are compact with xy 5 q, then

either x 5 q or yn s q for some n E Z In the example of
n
a lattice-ordered group, C= C for each n, so primary is

equivalent to prime.



2.6. Lemma: If q is primary and p = rad(q), then p is

prime.



Proof: Let x and y be compact, with xy 5 p and y 4 p.

Since xy is compact, by our proof of Proposition 1.11 we

have (xy)n <_ q for some n E Z and we assume n is the

smallest positive integer with this property. If n = 1,






15
k +
xy q, so x < q < p or y _< q for some k E Z But y $ p,
k +
so y k q for any k E Z so x < p, and we are done. If

n > 2, then (xy) n = (xy)n- xy < q, so again since yk q

for any k E Z (xy) n- x <- q. But by choice of n,

(xy) n-I q, so xm < q for some m E Z and x < rad(q) = p.D



This justifies the following terminology: If q E L is

primary and p is prime with p = rad(q), then q is p-primary,

and p is the prime associated with q.

Note: If q is p-primary, and x and y are compact, with

xy < q, then x < q or y s p; i.e., one of x and y is below q,

or they both lie below p, since L is commutative.



2.7. Lemma: If q is p-primary in L, and a $ p, then

[q:a] = q.



Proof: As always, q < [q:a]. If x is compact with

ax q, then since a $ p, there is y compact with y < a

and y $ p, and so xy < q. Now, y $ p, so x < q, and so

q > v{x compactlax < q} = [q:a].D



2.8. Lemma. If p,q E L satisfy the conditions:

(1) p > q

(2) If x is compact and x < p, then xn < q for some

n E Z+

(3) If x and y are compact and xy 5 q, then x q or

y < p
then p is prime and q is p-primary.






16

Proof: If x and y are compact and xy q, then by (3),

x : q or y 5 p, so by (2), x q or yn q for some n E Z+J,

and q is primary. Next, by (2), p rad(q), by Proposition

1.11. Now, if x is compact and xm q for some m E ,

assume m is the least positive integer with this property.

If m = 1, we are done by (1). If m 2, then xm = xm-lx q,

but xm-1 is compact and xm-1 $ q, so x p by (3). Thus

p = rad(q) and so p is prime.



2.9. Proposition: If p is prime and ql,q2,... ,qn are
n
all p-primary, then A q. is p-primary.
i=l 1


n
Proof: Let q = A q.. First, p > ql, so p a q. If x
i=l
is compact and x p, then for each integer i with 1 < i n,
m.
there is a positive integer m. with x < q., so if

m = max{mil
with xy s q and y $ p. Then xy s qi for each i, and so

x < qi for each i, 1 i s n. Hence x 5 q, and q is

p-primary.0



2.10. Proposition: If q E L is p-primary and a t q,

then [q:a] is p-primary.



Proof: Let x0 be compact, with x0 < [q:a]. Since

a $ q, there is a y0 compact with y0 a and y0 t q. Now,

Y0X0 < a[q:a] q, so since y0 $ q, x0 s p. Thus








[q:a] 5 p. If x is compact and x < p, then xn < q < [q:a]

for some n E Z. If x and y are compact and xy 5 [q:a],

with y $ p, then for each z compact with z a,

(zx)y axy < q, so zx s q. Thus ax : q, and x 5 [q:a].

By Lemma 2.8, [q:a] is p-primary.D



2.11. Proposition: If {p.illi < n} is a finite set of
primes in L, and if for each integer i between 1 and n, qi
n
is a p.-primary, and if a = A q, then each prime p a
1 i=l
n
exceeds some pi (1 : i 5 n), and so rad(a) = A pi.
i=l


n n
Proof: First, II qi A q. 5 p, so for some i with
i=l i=l

1 < i 5 n, qi 5 p, since p is prime. Hence pi = A{primes

exceeding q.} p.D



We will say that an element a has a primary decomposition
n
if a = A q., where each q. is primary. For each i between
i=l
1 and n, let pi = rad(qi). If a has a primary decomposition
n
a = A q., and if for each i and j with 1 i n and
i=l

1 j < n, i j implies pi p., and if for each i with
n
1 i n, q. i A qk' then we call a = A q. a normal
1 ki i=l
primary decomposition of a. A refinement of a primary
n
decomposition a = A q. is a primary decomposition
i=l
n
a = A r., where for each j with 1 5 j s m, there is an
j=l 3








integer ij, 1 < i. n, with r. j q. In light of
3
J J J 1 .
n
Proposition 2.9, if a = A q. is any primary decomposition
i=l 1
of a, we can group together those qi's that have the same

radical, and their meet is again a primary with the same
m
radical, so we obtain a = A r., where if j, j,2' then
j=l 1 2
rad(r.j ) ;rad(r.j2 ), and each r. q. for some i. Now, if

for some J0, 1 < J0 < m, r. 2 Aj r., then we can exclude
30 j ;C

r. from the decomposition, and so we see that every primary
3O

decomposition has a normal refinement.


m n
2.12. Theorem: If a = A q. and a = A r. are two normal
i=l j=l 3
primary decompositions for a E L, where each qi is p.-primary

and each r. is tj-primary, then m = n and, after suitable

rearrangement, pi = t. for each i (1 i 5 m).



Proof: If a = 1, then m = n = 0, and we are done. If

a < 1, then from {pl,...,p } u {tl,...,tn}, we choose an

element not strictly below any other. We may assume without

loss of generality that after rearranging terms of the

decompositions, we have chosen p m Suppose p {tl' f t }t .

Then qm $ t. for any 1 j n. By Lemma 2.7, [r. :qm] = r.
n n
for each 1 5 j : n, so [a:qm] = A [r.:q ] = A r. = a.
=j=l j= 3

Also, pm $ pi if 1 < i m-1, so qm $ pi for each 1 < i m-1,

and Eqi:qm3 = qi for 1 i s m-l. But [qm:qm] = 1, so







m m-i m-i
a = [a:qm] = A [qi:qm] = A qi, and qm A qi, contradicting
i=i i=l i=l
m
the assumption that a = A q. is a normal decomposition (if
i=l
m = 1, then a = [a:q ] = 1, a contradiction). Thus

Pm e {t ,...,t n}, and we may assume that with a suitable

rearrangement, Pm = tn. Let q = qm A r By Proposition

2.9, q is p -primary, and [q.:q] = q. if i s m-1, and
m-i n-i
[r.:q] = r. if j n-1. Thus [a:q] = A qi = A rj, and both
J J i=l j=i

are normal decompositions. Hence we may continue the process,

and we need only show that m = n. Suppose m < n. After
n-m n-m
m steps, Pm = tn'...,p = tn l and 1 = ^ r. A t. < 1
tn 1 nrn+i'j=1 j=1 I

since t, prime. This is impossible, and so m = n.0



We now proceed to the primary decomposition. As stated,

we will assume in Theorem 2.14 that each element of L is the

join of a set of compact, power-multiplicative elements, and

we will also assume that L is a modular lattice; i.e., for all

a,b,c E L, if a b, then a A (b v c) = b v (a A c). Again,

the motivating examples satisfy this condition, although we

only use it to show the existence of a decomposition, and

we are not certain that it is needed there.

We say that an element a E L is A-prime (or finite-meet

irreducible, or irreducible) if a = b A c implies that

a = b or a = c.



2.13. Lemma: If L is a mi-lattice with ACC, then each
n
a E L can be written as a = A a., where each a. is A-prime.
i=l 1








Proof: Let S = (a E LI a has no decomposition as
n
a = A a., with each a. an A-prime}. The lemma states that
i=l 1
S = P; we suppose not. By the ACC, there is an element a

which is maximal with respect to S. But then a = b A c,

with a < b and a < c, and so b J S and c J S. Thus
m n
a = ( A b.) A ( A c.), where each b. and each c. is A-prime.
i=l j=l 3 1
This is impossible, and so S = .D



2.14. Theorem: Let L be a modular, commutative mi-

lattice in which each element of L is the join of a set of

power-multiplicative elements. If L has ACC, then each

element of L has a normal primary decomposition, which is

unique up to order and the set of radicals of the primaries.



Proof: In light of Lemma 2.13, Theorem 2.12, and the

fact that every primary decomposition has a normal refinement,

we need only show that if a is A-prime, then a is primary.

Suppose a E L is not primary. Then there are compact elements

n +
b and c with bc : a but c $ a and b a for each n E Z.

We may assume that b is power-multiplicative, for b = v bi,
idI
where each bi is compact and power-multiplicative, and since
m n +
b is compact, b = v b Since b $ a for each n E Z
k=l k
there must be one of these b.k 's, which we denote b, such

n +
that b0 $ a for each n E Z Then b0c s bc < a, and we may

replace b with b0. Now, bc a, so [a:b] a v c > a. Also,

[a:bn] [[a:bn]:b] = [a:b n+1], so we have the ascending








chain a < [a:b] [a:b2] < .... This chain must break off

by the ACC; i.e., there is a n E Z+ with [a:b] I = [a:b] I

for each k > n. Now, [a:bn I > a, and also since bn $ a,
n n n
we have a < a v bn. Thus a < [a:b ] A (a v bn). Let

d = [a:bn] A (a v bn). Then d < [a:bn], so dbn < a. Now,

bn is multiplicative, so if r = [(d A bn ):b n], then
n n
b r = d A b. Thus, since d > a, we have

a v b r = a v (d A b ) = d A (a v b ) = d, by modularity.

Hence a bn d = bn a v b2n r, so b2nr a, so

r : [a:b2n] = Ca:bn], so bnr < a, and d = a v bnr = a. Thus

a is not A-prime, and so every A-prime element is primary.0














CHAPTER III
MULTIPLICATIVE SUBSETS, LATTICES OF
EXTENSIONS AND LOCALIZATIONS

We now turn to the idea of localizations, which play an

important role in valuation rings and Priifer domains, as well

as in algebraic geometry. A localization is a special type

of the more general object known as a ring of quotients. The

lattice of extended ideals in a ring of quotients preserves

most of the essential information below a given set of primes

in the lattice of ideals of the original ring, including

prime and primary elements, joins, meets, products, and

compactness. In a commutative ring with identity, each ideal

A is the intersection of the set {AI|X E A}, where A. is the

contraction of the extension of A relative to the maximal

ideal MX (A is the indexing set for the set of maximal

ideals), and it is this fact that makes much of the ideal

theory of valuation rings and Priifer domains work.



3.1. Definition: A nonempty subset M of a mi-lattice

L is called a multiplicative subset of L if each m E M is

compact, 0 4 M, and whenever m1 and m2 E M, we must have

m mm2 E M.

One way of constructing a multiplicative subset of a

mi-lattice L is to take {p,|X E A}, a nonempty set of

pairwise incomparable primes, and take a subset M0 of








compact elements with the properties:

(1) For each X E A and m E M0, m $ pX.

(2) If q is a prime and, for each X E A, q $ p., then

there is a m E M0 with m 5 q.
n
Set M = { m I n 1 and for each i, 1 i n, mi E M0}.
i=l
M0 S M, so M (. Each element of M is compact, since each

element of M0 is. Since {pi| E Al 4, let p be an element.

Since 0 p, no finite products of elements of M0 can be 0,

for no m E M0 satisfies m : p, and p is prime. Thus 0 4 M,

and clearly any finite product of elements of M is again in

M. The fact that every multiplicative subset of L arises in

this fashion is the content of Lemma 1.10. Recall the nota-

tion that if a E L and for each m E M, m 4 a, then

Sa = {b E Lia b and for each m E M, m 4 b}. Notice that

if a1 a2, then S a Sa so that each Sa is contained in
j. &a1 a2 a
S0. We let {p, | E A} be the set of elements maximal in S0.

This is a nonempty pairwise incomparable set of primes, and

if b E L such that for each A E A we have b $ p., then there

must be a m E M with m b. We say that {piX E A} is the

set of primes associated with M (or the set of associated

primes of M).

If M is a multiplicative subset of L, and {pIX E A}

is the set of associated primes of M, then let M = {m E Lf

m is compact and for each X e A, m t p,}. M c M and

0 4 Ms, so since Ms is closed under multiplication, it is a

multiplicative subset of L containing M, with the same set

of associated primes. M is called the saturation of M,
S5i"""" '" -








and we say that a multiplicative subset M of L is saturated

if M = M s. Clearly, Ms is saturated, and it contains every

multiplicative subset of L which has {p|IX E A} as its set

of associated primes.

In commutative ring theory, if R is a commutative ring

and M c R is a multiplicative set not containing zero, then

when the ring of quotients is formed, all ideals of RM are

extended ideals; i.e., they are all ideals generated by the

subset of RM corresponding to an ideal of R under the

canonical homomorphism D of R into RM. Also, each ideal of

RM meets the image of R in an ideal of that subring, and the

inverse image of these intersections are called contracted

ideals. This is all put forth in Gilmer [3], and it is

shown that all extended ideals are of the form


Be = {$(b)/U(m) I b E B and m E M}


while all contracted ideals are of the form


Bec = {x E Rxm e B for some m E M}.


With this in mind as a motivation, we proceed to our next

definition.


3.2. Definition: Let M be a multiplicative subset of L,

and let {p,[X E Al be its set of associated primes. If

a E L, then set








*
a (M)= v{x compact xm < a for some m E M}.


*
Notice that a (M) > a.




3.3. Lemma: If a E L and x is compact, then
*
x < a (M) xm < a for some m E M.




Proof: Certainly if xm < a for some m E M, and x is
*
compact, then x < a (M). Conversely, suppose x is compact and
*
x < a (M). Then x < v{y compact ym < a for some m E M}, so
n
since x is compact, x v Yk' where for each k < n, Yk is
k=l
n
compact and there is a mk E M with ykmk < a. If m = II m,
k=l
n
then m E M and m < mk for each k < n, so xm < v Ykmk < a.0
k=l



*
3.4. Lemma: a (M) = a (M ).



*
Proof: Since M c Ms, we have a (M) < a (M ). But if x

is compact and x 5 a (Ms), then there is a m E Ms with xm a.

Now, for each X E A, m $ p,, so by the discussion following

Definition 3.1, there must be a m0 E M with m0 < m. Hence
*
xm0 < xm < a, so x < a (M), and a (M) = a (Ms).D



When it is clear from the context just which multipli-

cative set we are referring to, we will write a* rather







*
than a (M). Further, we may always assume that we are using

a saturated multiplicative subset. This has no effect upon

our results, but it is convenient, and by Lemma 3.4, it is

justified.



3.5. Definition: If M is a multiplicative subset of L,

we define an equivalence relation on L by:


a ~ b 4> a* = b*.


(If it is necessary to refer to M, write a -M b.) Let a

represent the equivalence class of a under ~((a)M, if

necessary). We will say a is the extension of a, and a* is

the contraction of a.

A /

3.6. Proposition: a* = v{blb E a}, a* E a, and

(a*)* = a*.



Proof: Since b < b* for all b e L, we have

a* v{blb e a}, and a* 5 (a*)*. Let x be compact with

x < (a*)*. Then there is an m1 e M with m1x < a*. But m1 is

compact, so m1x is compact, and there must be a m2 e M with

m2mlx 5 a. m1m2 E M, so x < a* and (a*)* = a*. Thus
a* ~ a, so a* e a, and a* = v{blb e a}.D



3.7. Definition: We now want to create a new mi-lattice

with these equivalence classes. To this end, we set


a 5 b 4> a* -< b*.





27
A
This is a partial order on {aj a E L}. We need it to be a

lattice order which gives us an algebraic lattice.



3.8. Lemma: Let a,b E L. The following are equivalent:

(1) a < b.

(2) There is an a0 E a with a0 < b.

(3) There is a b0 E b with a < b0.

A% A A%
Proof: (1) => (3): If a < b, then a < a* < b* E b.

(3) => (): Suppose b0 E b, with a < b0. Let x be

compact, x 5 a*. Then there is a m E M with mx < a, so

mx 5 b0, and x < b* = b*, so a* < b*, and a < b.

(1) => (2): If a < b, then a* -< b*, so for each x com-

pact with x -< a*, we choose mx E M with m x < b. Let

a0 = v{m xjxi compact, x < a*}. Since a0 < b, we only need

0 E: a; i.e., a* = a*. First, a* < a* by choice of a0. Let

y be compact, y _< a*. Then there is m E M with

my a0 = v{m xx compact, x < a*}. Now, my is compact, so
n
my < v mkxk, where for each k < n, xk is compact, xk < a*,
k=l
and mk = mk as chosen above. But since xk < a*, mkxk < a*
Xk

with mkxk compact, so we can find pk E M with Pkmkxk < a.
n n n
Let m0= m HI k. We have m0 E M, and m0y < ( H "k) ( v mkX) <
k=l k=l k=l k k
n
v (kmkXk) < a, and so y < a*. Thus a* = a*, so a0 E a.
k=1
A
(2) => (1): Suppose a0 e a with a0 < b. If x is compact

and x < a*, then there is a m E M with mx a0 < b, so x < b*.
A A
Thus a* < b*, and a < b.









3.9. Proposition: If {a.ili E I} is any family in L,
A
let a = v a.. Then a = v a., and so ( v a.)* v (at).
iEl iel 1 iEI iEI



Proof: First, since a a. for each i E I, we have
A A A As
a ai for each i E I. Suppose c 2 a. for each i E I. Then
1^
for each i E I, there is b. E ai with c b.. Thus

v b. c, so if b= v b., b < c, and we need to show that
iel 1 iEI

b = a. If x is compact and x 5 a*, then there is m E M with

mx a. Thus mx s v{y compactly < a. for some i E I}, so

n
since mx is compact, mx < v Yk' Yk compact and for some
k=l
ik E I, Y ai a? For each k : n, choose mk E M with
kk 1k k
n
mikyk < b. Then (mm1m2...m n) E M and mml...m X v Ykmk <
lkn k=l
n
v b. v b., so x b*, and a* b*. A similar argument
k=l k ieI

shows that b* < a*, so a = b < c.D



Thus joins are preserved going from L to {ala E L}, and

{ala E L} is closed under (finite and infinite) joins. The

next result yields the fact that we have constructed a

complete lattice.



3.10. Proposition: If {aili E I} is a family in L, let
A A A
a = A (at). Then a = A a., and a = a*, so {vlv E L} is
iel 1 ieI
complete. Further, if r and s are in L, with t = r A s,

then t = r A s and t* = r* A s*.






29

A A A
Proof: First, a : a* E a. for each i E I, so a a.
1 1 1
for each i E I. Suppose c a. for each i E I. Then c at
11
-' A A
for each i E I, so c a. Thus c a, and a = A a.. Now,
iEl 1
if x is compact and x 5 a*, then for some m E M,

mx a= A at, so mx < at for each i E I. But then for
iel 1

each i E I, there is m. E M with m.mx ai, since mx is

compact, and so x at for each i E I since m.m E M. Hence
1 1
A A
x a, so a = a*. Let t = r A s, and let d E r A s. By

what we have just shown, d* = r* A s*. If x is compact and

x r* A s*, then there are elements m and m2 in M with

m1x : r and m2x s. Thus m1m2 E M with mlm2x : r A s = t,


so d* = r* A s* s t*, and since t 5 r* A s*, we have
A A A,
t* = r* A s* = d*, and t = r A s.[


A A^
3.11. Lemma: For each a E L, a is compact in {blb E LI =

there is an element a0 E a with a0 compact in L.


A /
Proof: If a0 is compact in L, then suppose a0 < A c..


By Proposition 3.10, a0 a* < A (c), so since a is
0 0 lEI 1
n
compact, a0 5 A c which we will label c. Then
k=l 'k
A A n ^ A^
a0 < c = A c. ,so a0 is compact in {blb E L}.
0 k=l 'k
A A
Conversely, if a is compact in {bjb E L}, then since L

is algebraic, a = v{xfx is compact and x a}. By Proposition
A ^A n ^
3.9, a = v{xlx is compact and x < a}. Thus a = v Xk, where
k=l








for each k < n, xk is compact and xk a. But then if
n
a0 = v xk, a0 is compact and again by Proposition 3.9,
k=l
^ n ^
a0 = v xk = a.0
k=l



Propositions 3.9 and 3.10 together with Lemma 3.11,
^
imply that {a[a E L} is an algebraic lattice, which we will

denote LM. We tie all of this together in the next result.



3.12. Theorem: If L is a mi-lattice and is a (satu-

rated) multiplicative subset of L, then LM = {ala e L} is an

algebraic lattice, with least element 0 and greatest element
A ^
1 = {b E L I there is some m E M with m < b}. Thus M c 1,

and 1 is compact.



Proof: Propositions 3.9 and 3.10 show that LM is a

complete lattice, and Proposition 3.9 and Lemma 3.11 show

that a = v{xlx is compact and x a}, so that LM is algebraic.
A^ A A,
0 < a 5 1 for each a E L, so 0 a < 1. To see that

1 = {b E Lib m for some m E M}, suppose b m for some

m E M. Then for each x compact in L, mx b, so b* = v{x E LI

x is compact} = 1. On the other hand, if b* = 1, then let

m 0 E M. Since m0 is compact and m0 5 1, there is a m1 E M

^
with m0m1 < b, and so we set m = m0m1 E M. Then 1 = {b E LI

b 2 m for some m E M}, and in particular, M E 1. Since

M e ( and each element of M is compact, 1 is compact.D








Now we must introduce a product on L There is a per-
AA A
fectly natural way to do this, namely ab = (ab), but we

need to check that this is well-defined.



3.13. Lemma: If a and b are in L, then ab ~ a*b*.



Proof: Clearly (ab)* < (a*b*)*, so we need only show

a*b* (ab)*. But a*b* = v{x1x21x1 and x2 are compact,

xI < a*, x2 s b*} = v{x1x21x1 and x2 are compact, and there

are ml,m2 E M with m1x1 a and m2x2 b} S V{x1x21x1 and x

are compact, and for some m E M, mx1x2 ab} s (ab)*.D



Thus if a = a2 and bI = b2, then albI ~ a*b* = a*b* ~

a2b2, so we define ab = ab.



3.14. Theorem: If L is a mi-lattice and M is a multi-

plicative subset of L, then LM is also a mi-lattice, in which
AA A A
1 is compact and 1a = a for each a E L.



Proof: In light of Theorem 3.12, we need only consider

the product on LM. It is both associative and commutative,

since the product on L is. Since ab 5 a A b for each a,b E L,

we have ab S a A b by Proposition 3.10. If a E L and

{bili E I} is a family in L, then let b = v b. andc=ab. Then
iEI
a( v b.) = ab = c= v ab. If x and y are compact in LM,
iEI iEI







A
then there are compact elements x0 E x and y0 E y, so that
A A A A
xy = x0Y0 is compact since x0y0 is compact. Since l*a 5 a,
A< A A
we have l-a a. But if x is compact and x < a*, then there
2 2
is a m E M with mx a. Now, m E M and m x s ma 5 la, so

a* < (l-a)*, and 1-a = a.D



The last part of Theorem 3.14 is an expected result if

this is to parallel commutative ring theory, for if R is a

commutative ring and if M is a multiplicative subset of R

with 0 4 M, then RM is a commutative ring with identity.

Now, the following result is part of an exercise in Gilmer

[3], and is easily verified.



3.15. Proposition: If S is a commutative ring, then S

has an identity if and only if SA = A for each ideal A of S

and S is finitely generated as a S-module.



Thus, for our version of a ring with identity, we let

L have the properties that 1 is compact and la = a for all

elements a E L.

Now, we would like this lattice of extensions LM to

reflect all of the information below the set of associated

primes of M, including prime and primary elements, radicals,

and residual quotients by compact elements. First, we

consider the primes.








3.16. Lemma: If {pXIA E A} is the set of primes
A
associated with the multiplicative subset M, then {pXIA E A}
A

is the set of maximals in LM. If X1 d X21 then P, and p2

are incomparable, and for each A E A, pX = p* and pA is prime

in LM.

^. -A

Proof: p. < 1 for each A E A, for if not, then for

some X E A, p, E 1, and p X m for some m E M, which is
A-

impossible. Also, if a E L and a > p, for some A E A, then

there is an a0 E a with a0 > p., which is maximal in

S0 = {b E LIfor each m E M, m b}. Thus there is a m E M
,% A A
with a m, so since M c 1, a = 1, and PA is maximal in LM.
As 2\ A\ A A\
Since (1) = 1, p. is prime for each X E A. If a 1, then

since M c 1, for each m E M, a t m. Thus a E So, so a p,

for some A E A, and a pX. This says {piX E A} is the

set of maximals of LM. Now, if XAX2 E A with A1 tX 2, then

p and p are incomparable, so let x1 and x2 be compact with
P1 P2

x 1 PXl x2 P2 x 1 p 2 and x2 $ p .l For each m E M,

2?1 2 2 1*
m pA2, so mxI x pA and mx2 $ pA Thus xI 1 pl and

x p* so p* and p* are incomparable, and so are P
2 X2 X1 X2 X1
A A\ A A
p2. If X E A, pX ; 1, so since p* E pX, p* does not exceed
2*
any m E M, and p* : p0 for some X0 E A. Thus pA : p? < p0
Xnd 00 X 0
an0 = since the p.'s are pairwise incomparable.0
and s PA 0








A
3.17. Theorem: The primes of LM are those classes p

where p is prime in L and p pX for some p. in the set
^0 ^0
{pX|X E A} of primes associated with M. If p is prime and

p p. for some X E A, then p = p* and there are no other

primes of L in p.


A A A
Proof: Suppose a is prime in L,. Then a < 1, so

a p, for some X E A, and a* < pO = pO If bc < a*,
0 0 X0 0
.A A\ A\ A\ A A A
then bc a, so b a or c : a, and so a* is prime since

b a* or c a*. If q E a with q prime in L, and if x is

compact with x < a*, then for some m e M, mx < q. But

q a* < p 0, so m $ q, and so x < q. Thus q = a*, and a*
0A

is the only prime in a. On the other hand, suppose p is

prime in L, with p < p. for some X E A. If ab = p, then

ab : p*. Suppose b $ p*. Then there is a x compact with

x $ p* but x 5 b. For each y compact with y 5 a, xy is

compact and xy < p*, so there is an element m E M with

mxy p. Now, p py so m $ p, and so xy < p. But x $ p*,
1
A A A\
so y 5 p, and so a p. Thus a p, so p is prime, and by the

first part of the proof, p = p* is the only prime in p.E



The theorem establishes a one-to-one correspondence

which preserves order between primes of LM and primes of L

that do not exceed any elements of M. We now want to do the

same for primary elements, but first we consider radicals.









3.18. Proposition: If a E L, and if r = rad(a), then
/% /A
r = rad(a) and r* = rad(a*).



A /%
n+
Proof: First, r = v{xl x is compact andxn 5 a for some E Z +}.
n /n A
If x is compact and x < a, then x is compact and (x) < a.
A A
Conversely, if (x) < a with x compact, there is x0 E x with

x compact in L. Now, n < a*, so sincexn is compact, there is
<0 a.0 T h

a m E M with mx _< a. Thus (mx)0Jn 5 mxn < a, mx0 is compact,
A/ A A A A A% A / j s opc ad%

and mx0 = lx0 = x0 = x, so r = v{xjx is compact and xn < a

+ A A
for n E Z } = v{xlx is compact and (x) < a for some

+ /
n E } = rad(a). If x is compact and x -< r*, then there is

m E M with mx < r, so for some n E Z +, (mx)n < a, and
AA y A // /A/A A% /A /A
(mx) < a. But mx = lx = x, so (x)n < a, and xn < a* with

x compact. Thus r* < rad(a*), but if rad(a*) = s, then
/% /% /%, /%
s = rad(a*) = rad a = r, so r* = rad(a*).D




3.19. Theorem: Let M be a multiplicative subset of L

with {pI| E A} the set of associated primes of M. Let p be

a prime in L, with p < p. for some X E A. If q is p-primary,

then q is p-primary in LM with q = q* the only primary in q,
/% /%
for p or any other prime. Conversely, if a is p-primary,
/%
then a* is p*-primary, and a* is the only primary in a.


/< /%
Proof: Suppose a is p-primary in LM, with p prime in L
A /A
and p < p. for some X E A. Then p = rad(a), and p = p* =

(rad(a))* = rad(a*). If x and y are compact with xy < a*,
t A A an y A A A /A a
then x and y are compact and xy < a, so x -< a or (y)n < a








Z+ n
for some n E Z Thus x a* or y < a*, so since a* < p < ,

a* is p-primary. If q E a with q primary, then if x is com-

pact and x < q*, there must be m E M with mx < q. But since
m ,mn frecn 7+ n
m E M, mn E M for each n E Z and so m n q. Thus x < q,

so q = q* = a*. Conversely, if q is p-primary in L, with

p -< p for some X E A, then q p., so q < p, < 1. Now,
/\ ^ A /\
p = rad(q), so p = rad(q). If x and y are compact with

xy q, then let x0 and y0 be compact in L with x0 e x,
A
Y0 e y. Since x0y0 q*, there is a m E M with mx0y0 q.
Z+ mn mn
Again, for each n E Z mn E M, so mn $ q. Thus x0y0 < q,
n +A A A
so x0 < q or y0 < q for some n E Z and so a = a+0 q or
A A ,
(b) = (b0) n q. Therefore q is p-primary and by the

first part, q* is p-primary and is the only primary in q, so

q = q*.D



Finally, we turn our attention to residual quotients.



3.20. Proposition: Let a,b E L. Then

(1) If d = [a:b], then d [a:bl.

(2) If b is compact and d = [a:b], then d = [a:b].

(3) ([a:b])* < [a*:b*].

(4) ([a*:b*])* = [a*:b*].

(5) If c = [a:b], then c* = [a*:b*].


A A. At AA^ A A A
Proof: (1) Ford=v{xlbx< a} < v{xjbx a} = [a:b].

(2) Let b be compact in L, and let x be compact with
bx A a. There is a x0 E x with x0 compact. Then bx0 is
bx a. There is a Xoe x with x0, compact. Then bx0. is








compact, with bx0 < a*, so for some m E M, mbx0 A a.
A A\ A A A A\ AS
mx,0 = lx = x, so x 5 v{yly is compact and by a a} = d, so

by part (1), d = [a:b].

(3) For b*[a:b]* <(b[a:b])*
(4) As always, [a*:b*] a ([a*:b*])*. Let c = ([a*:b*3)*.

Now c= v{xx compactand xm [a*:b*] for some m e M} =

v{xjx compact and b*xm a a* for some m E M}. If x is compact

and b*xm a a* for some m e M, then for any y compact with
A A\ A A A\ A
y a b*, we have yxm a a*. Thus yxm = yx < a, so yx < a*

so b*x a a*. Therefore, c = v{xlx compact and b*xm s a* for

some m E M} 5 v{xlx compact and b*x 5 a*}, so c = [a*:b*].

(5) Let c = [a:b] = v{ylby o a}. If x is compact with

x < c*, then x < c, so bx < a, and b*x* < (bx)* < a*. Thus

x a x* A [a*:b*], so c* a [a*:b*]. Conversely, if x is com-
A\A A\
pact and x < [a*:b*], then b*x < a*, so bx < a. Thus
A. A A\ A
x a [a:b] = c, so x < c* and c* = [a*:b*].D



3.21. Definition: If p is prime in L and M = {mlm is

compact and m I p}, we say that LM is the localization at p,
P
p
and use the notation L = LM .
p

In commutative ring theory, if R is a commutative ring

with 1, then each ideal Ais the intersection of the contractions

of all of its extensions in the localizations at the maximals

exceeding A. This allows one to assume a unique maximal ideal

when trying to establish an arithmetic result, since by

passing to the localization, all finite joins, meets, and

products are preserved. We obtain the same result here.








3.22. Theorem: If L is a mi-lattice with 1 compact and

1la = a for each a E L, then let {p. i E I} be the set of
1
maximals of L. For a E L, let (a)i be the class of a in

L i, and let a. = v{blb E (a).} = a* with respect to Li

Let J = {j E I la p.}. Then a = A a. = A a..
J iEI 1 jEJ 3



Proof: Since a < a. for each i < I, a A a. < A a..
1 iEI i jEJ 3

Let x be compact with x < A a.. For each j E J, x < a., so
jej 3

there is a compact element m. with m. $ p. and m.x a.

Then [a:x] 2> mi, so for each j E J, [a:x] 4 p.. Since 1 is

compact and a < [a:x], [a:x] = 1. Thus x = 1-x < a, so

A a. :_ a.0i
jEJ 3














CHAPTER IV
INVERTIBLE ELEMENTS

We now consider the analogue of invertible ideals of

commutative ring theory. Recall that in a commutative ring

R, if an ideal A contains a regular element (an element that

is not a zero-divisor), then A is invertible if and only if

it satisfies two properties:

(1) A[B:A] = B for each ideal B c A.

(2) [AB:A] = B for each ideal B of R.

An ideal satisfying (1) is multiplicative, while an ideal

satisfying (2) is cancellative. We have already mentioned

that every principal ideal, and thus every power of a princi-

pal ideal, is multiplicative. If a principal ideal is

generated by a regular element, then the ideal is cancellative

as well, and thus is invertible. For these reasons, when we

consider properties of principal ideals or elements of the

ring, it is often enough to look at multiplicative and invert-

ible compact elements. One thing should be mentioned here:

when dealing with the lattice of convex Z-subgroups of a

lattice-ordered group, if a < 1, then 1-a = a-a = a since

multiplication is intersection, and so one will never en-

counter any elements besides 1 that could be called

cancellative or invertible in the sense of (1) and (2) above.






40

4.1. Definition: If L is a commutative mi-lattice, an

element a E L is cancellative if [ab:a] = b for each b E L;

equivalently, if ab = ac, then b = c, so that multiplication

by a is an injection. An element a E L is invertible if a

is both cancellative and multiplicative (Definition 2.3);

i.e., multiplication by a is an order-preserving bijection

of L onto [0,a] = {b e LJ0 b a}. We now list some

results concerning the nice properties and behavior of

invertible elements.



4.2. Corollary: If a is invertible and b is multiplica-

tive, then ab is multiplicative.



Proof: Since [ab:a] = b, the result follows from

Lemma 2.4.D


n
4.3. Lemma: Let a = I a.. Then a is cancellative =>
i=l
each a. is cancellative.
1


Proof: If a is cancellative, let i0 n. For each

b E B, [ab:a] = b, so b = [ab:a] = [[ab:a. i]: ii a.] i
10 i i0

[(( n a.)[a. ib:a. 1]): Ili a. ] [a. b:a. i] b, so
imi 0 0 0 ii 0 0 0

b = [ai0 b:a. ], and each a.i is cancellative. Conversely,


suppose al,...,an are cancellative. We proceed by induction

on n. If n = 1, a = a1 is cancellative. Suppose that for








each positive integer k : n-1, if C,1...,ck are cancellative,
k
then II c. is cancellative. Then for b E L,
j=l n-i n-i n-i n-i
[ba:a] = [[((b a.)an) :an ]: II a.] = [(b H a.) : n a.] = b
i=l1 i=li1 i=1 i=li1
by induction, so a is cancellative.0


n
4.4. Proposition: If a = H a., then a is invertible
i=l 1

Each a. is invertible.
1


n
Proof: If a = n a. is invertible, then a is cancella-
i=l 1

tive, so by Lemma 4.3, each a. is cancellative. If n = 1,
1
we are done, so let n > 1. Let i0 5 n, and suppose b = ai .
10

Then b H a. : a, and b = [(b H a.): I a.] by Lemma 4.3.
i 1i0 i .i0 i ei0
i^i .O 1 ^O1

Thus a. [b:a. ] = E(a[b:a. ]): I a.] since H a. is can-
1 ii 1 1~i 1
1i0 0 0 1

cellative, and so a. i[b:a. i = [(a[[(b IT a.): H a.]:a. ]) :
10 10 0 1 1

Sa.] = [(a[(b n a.) :a]): n a. ] = [(b H a.) : IT a.] =b,
i^i i0 ii0 ii0 i0i0 ii0

so each a. is multiplicative, and thus invertible. Converse-
10
ly, if al,...,a are invertible elements of L, then by Lemma
n
4.3, a = R a. is cancellative. We proceed by induction on n.
i=l 1
If n = 1, a = a1 is invertible. Suppose that for each

positive integer k n-l, whenever cl,...,ck are invertible
k n-i
elements we must have I c. invertible. Then H a. is
j=l 3 i=l 1
invertible and a is multiplicative, so by Corollary 4.2, a

is invertible.0








4.5. Proposition: If any a E L is invertible, then for

each b E L, l-b = b, so that 1 is invertible.



Proof: Let a be invertible. Then a is multiplicative,

so a = a[a:a] = a*l. Then b-l = [(b-l-a):a] = [b-a:al = b.0



4.6. Proposition: If a is invertible, then

a( A bi) = A abi.
iEI idI



Proof: First, A b. = A [b.a:a] = [( A b.a):a] by
iEI iEII iEI
by Proposition 2.2, part (f), so a( A b.) = a[( A bia):a] =
iEI iEI
A (b.a) since A b.a < a.0
iEI i 1I
1e ielI


This result states that if a is invertible in L, then

multiplication by a is a lattice isomorphism of L onto [0,a]

which preserves meets and joins of infinite as well as finite

families.



n
4.7. Proposition: If a is invertible and a = a for

some n 2, then a = 1.



Proof: First, l.a = a = an = (a n-l)a, and a is cancel-

lative, so an-l = 1. Then n 2 implies a 2 an- = 1, so

a = l.0








4.8. Lemma: If a is invertible and {a.ii I} is a

family in L with a. -< a for each i E I, then

( v a.) :a] = v [a :a].
iEI iEI



Proof: First, a[( v a.):a] = v a. = v (a[a.:a]) =
iEI iEI ieI

a(v [a.:a]). Then [( v a.):a] = v [a. :a] since a is
i EI iEI iEI
cancellative.



4.9. Proposition: Let L contain an invertible element.

Then the following conditions are equivalent:

(1) 1 is compact.

(2) Every invertible element in L is compact.

(3) There exists an invertible element a E L with a

compact.



Proof: (1) => (2): Let b be any invertible element of

L. Since L is algebraic, b = v b., where each b. is compact.
iEI 1

Then 1 = [b:b] = [( v b.):b] = v [bi:b]. Since 1 is compact,
iEI iEI
n
1 = v [b. :b]. Then b = 1-b = b( v [b. :b]) =
k=l k k=l k
n n
v (b[b. i:b]) = V b. and so b is compact.
k=l k k=l k

(2) = (3): Obvious since L contains an invertible

element.








(3) => (): Let a be compact and invertible in L, and

1 = v c.. Then a = a*l = v aci, so since a is compact,
iEI iEI
n n
a = v ac. = a v c. Since a is cancellative and a1l = a,
k=l lk k=l k
n
1 = v c. so 1 is compact.0
k=l k



In the theory of commutative rings there are several

conditions that are equivalent to the condition that a ring

R is an integral domain. We now consider some of these in

the context of mi-lattices. From the theory of commutative

rings with identity, the following conditions are equivalent:

(a) 0 (as an ideal) is prime.

(b) Every nonzero ideal exceeds a cancellative ideal.

(c) Every nonzero ideal exceeds an invertible ideal.

(d) Every nonzero ideal is the join of invertible

principal ideals.

In the language of mi-lattices, L is a mi-lattice in which 1

is compact, invertible, and:

(1) 0 is prime.

(2) Every nonzero element of L exceeds a cancellative

element.

(3) Every nonzero element of L exceeds an invertible

element.

(4) Every nonzero element of L is the join of compact

invertible elements.

Of course, the implications (4) => (3) => (2) => () are obvious.








We give examples to show that no other implications hold

in general, even if we assume ACC and modularity.

(1) # (2): L = {0,x,l}, 0 < x < 1, with x = x for

all n E Z and 1-a = a for all a E L. Here, 0 is prime,

since if a > 0 and b > 0, then a x and b x,, so
2
ab x = x > 0. However, 0 is never cancellative, and

since x-x = x-l, x is not cancellative, so x does not exceed

a cancellative element. See Figure 4.11.

(2) # (3): L = {0} u {pnqmlm,n 0} where p = q = 1,
n m n m rqs
1-x = x for all x E L, 0 = ^ p = ^ q and p q > pq <=>
n=l m=l
m + n < r + s or m + n = r + s n + s.

Here, l,p, and q are cancellative, and so by Lemma 4.3,

every nonzero element of L is cancellative. However,
2 2 2
p[q:p] = p q, and q[p :q] = pq < p so neither p nor q is

invertible, and so no element below 1 can be invertible, in

light of Proposition 4.4. See Figure 4.12.

(3) # (4): Let 1 = {pk k E Z, k 0} u tr}, with

Pk p* P if k Z Z, and Pk e r for all k E Z, k > 0. Order
by setting Pk pt if k < Z, with r = ^ P Set pk*PZ = Pk+'
k=l
2 2 {k
and pk r = rpk = r = r. Let = {qIk E Z, k 2 0} u {s},

with order given by qk > qt if k < , and s = A qk Set
k=0
2
qk'q = qk+k and q ks = s-qk = = s. Then
.4-
1i 2 = {(a,b)la E Q1, b E Q2} with (al,b1) < (a2,b2) 2

(b1 < b2) or (b1 = b2 and a1 a2), and (al,b1)-(a2,b2) =

(a1a2,blb2). Let L = [(P0,S)),(p0,q0)3 = {(a,b) E Xi Q 221

(a,b) (P0,S)}, with the inherited order and








(a1a2,b1b2) if bI 1 s and b2 s
(al,bl)'(a2,b2) =
(P0,S) if b = s or b = s.

Relabel as follows: (p0,q0) = 1, (pq0) = pk (r,q0) = r,
k kYk
POqk ) qk (k) = p q (pos) = 0, (r,qk) = rq. Then

multiplication is done in the obvious commutativee) way, with

1-x = x for all x E L, pkr = r = r2, x-0 = 0 for all x E L,

and p = q = 1. L is a mi-lattice with ACC; it is totally

ordered, so it is modular. Now, p is not multiplicative,

since p[q:p] = pq < q, so p is not invertible. But if

p = V a. with each a. E L, then p = a. for some i0 E I, and
iEI 1
so p is not the join of a family of invertible elements.

However, q is invertible, so q is invertible for each k > 1,

and every element except 0 exceeds some power of q. See

Figure 4.13.

Thus, we have many possible choices for a generalization

of the lattice of ideals of an integral domain. We choose

the strongest, since it is used the most.



4.10. Definition: A mi-lattice L is a domain lattice

if 1 is compact and each nonzero element is the join of a

set of invertible elements (which are compact by Proposition

4.9).



























0


FIGURE 4.11


co
n
A q=
n=l


1

p
q,
2
p
pq
2
q
3
p
2
P q
pq2
3
q
4
p
3
p q
2 2
P q
3
pq
4
q
5
p


00
n
A p =
n=l


00o
n
r = A p = pr'
n=l q

pq
Pq

2
P q
3
p q


qr
2
q
2
Pq
2 2
P q
3 3
p q





2r
g r4


FIGURE 4.12


0o
n
A qg
n=l


FIGURE 4.13


= 04


x I














CHAPTER V
GENERALIZATIONS OF VALUATION RINGS AND PREFER DOMAINS

In commutative ring theory, a valuation ring D is an

integral domain with identity in which the set of ideals is

totally ordered. This motivates our next definition.



5.1. Definition: A v-lattice is a domain lattice L

which is totally ordered. The following theorem lists some

basic properties of valuation rings that carry over to

v-lattices.



5.2. Theorem: If L is a v-lattice, let a E L with a < 1.

Then

(1) If a e 0 and a is compact, then a is invertible.

(2) Rad(a) is prime in L.
n k k+l
(3) If p0 = A a then p0 is prime. If a =a
n=l
for some k 1, then a = a, and a is prime.
no
(4) If p is prime and p < a, then p < p0 = A a .
n=l
(5) If b E L and a < rad(b), then ak b for some

k E Z+.



Proof: (1): If a t 0 and a is compact, then write
n
a = v a. where each a. is invertible. Then a = v a. since
iEI k=l k
a is compact, so that a = a. since L is totally ordered, and
n0
a is invertible.








(2): Rad(a) = A{plp is prime and a < p}. But

{pjp is prime and a < p} is totally ordered, so by 1.7 of

Keimel [5], rad(a) is prime.

(3): Let x and y be compact elements with x 4 p0 and
+ mn
y J po. Then for some m and n E Z x $ am and y a Thus

am < x and a < y, and since y is compact and non-zero, it

is invertible by (1). Hence amy < xy, and am+n _< amy, so

am+n < xy, and xy $ P0. Therefore, p0 is prime. Now, if
k k+l k
a = a for some k > 1, then p0 = a so that a = p0 and

a is prime.

(4): If p is prime and p < a, then p < a for each

k E Z so p < P0.

(5): If for each k E Z +, b t ak then b < ak for each
+ n k
k E Z so b : p0 = A q and rad(b) < p0 < a. Thus
k=l
a $ rad(b).L



5.3. Definition: If L is a mi-lattice and p is a prime,

then p is a branched prime if there is a p-primary q e p. We

say p is unbranched if p is the only p-primary in L. As for

valuation rings we have:



5.4. Theorem: If L is a v-lattice and p is a non-zero

prime of L, then let S = {qXIX E A} be the set of p-primaries.

(1) If q is p-primary and if x is compact with x $ p,

then q = q-x. If q is compact, then p is maximal.








(2) S is closed under multiplication, so

{pk k E Z+} < S. If p p2 then S = {pkk E Z+.

(3) If p is branched and q f p with q p-primary, then
CO
A qn = A{q|lX E A}.
n=l
(4) p0 = A{qXIX E A} is prime, and there are no primes

a satisfying p0 < a < p.



Proof: (1) q p < x, and since x is non-zero and

compact, x is invertible, so q = ax. q is p-primary and

x p, so a q, and thus a = q, so q = qx. If q is compact,

then q is invertible since q r 0, and if x is compact with

x J p, then xq = l-q = q, so x = 1, and p is maximal.

(2): If q, and q2 E S, then rad(qlq2) = p. If x and y
_n +
are compact with xy < qlq2, and if xn $ p for each n E Z,

then q, = qlx by part (1), so xy < qlq2 = xqlq2. Thus qlq2
2
is p-primary. Now, suppose p e p and let q be a p-primary.
2 2
Then p < rad(q), so q exceeds a power of p and hence a

power of p, by Theorem 5.2, part (5). Let n be the least

positive integer for which pn < q. If n = 1, then p = q. If
n-1 n-I
n 2 2, then since p > q let y be compact with q < y 5 p

q < y, so since y is invertible, q = ay. Since q is p-primary

and q < y, a p, so q = ay py < n <_ q, and q = p .

(3): Clearly A q A q" Now, q < p = rad(q.) for
n=l XEA
n +
each A E A, so for each X E A, q < q. for some n E Z Thus
^ qAn = ^{qjX E A}.
n=l








(4): If p is unbranched, we are done. If p is
CO n
branched, p0 = A q for each p-primary q < p, so by
n=l
Theorem 5.2 part (3), p0 is prime and each prime a < p

satisfies a p0.D



5.5. Theorem: If L is a v-lattice and p is a non-zero

prime, then the following are equivalent:

(1) p is branched.

(2) There is an element a t p with rad(a) = p.

(3) There is an invertible element x with rad(x) = p.

(4) p > v{blb is prime and b < p}.

(5) There is a prime r < p with no primes a satisfying

r < a < p.



Proof: (1) => (2): obvious.

(2) => (3): Choose x invertible with a < x p. This

can be done since a < p, each non-zero element of L is a join

of invertibles, and L is totally ordered. Then rad(x) = p.

(3) => (4): If x is invertible with rad(x) = p, then

x > b for each prime b < p, so x 2 v{blb is prime and b < p}.

But x is invertible and 1 is compact, so x is compact, and
n
so if x = v{blb is prime and b < p}, then x = v bk, and
k=l
since L is totally ordered, x = b0, a prime strictly below p,

and rad(x) e p, contradicting our choice of x.

(4) => (5): Let r = v{blb < p and b prime}. If x and y

are compact with xy 5 r, then since xy is compact,








n
xy i v bk, and since L is totally ordered, xy b0, where
k=l

b0 < p and b0 prime. Thus x < b0 or y : b0, so x : r or

y 5 r, and r is prime.

(5) => (1): We proceed by considering the localization

at p. First, if x is invertible in L, we show that x is

invertible in L If b < x, let b0 E b with b0 < x, and set
p 0
d = [b0:x]. Since x is compact, d = [b0:x] by Proposition

3.20, part (2). Thus x [b0:x] = xd = b0, and x is multipli-
A A\ As A* A\ A
cative. Now, suppose that xb = xc, with b,c E L Then for

each y compact with y :b, xy xc, so since xy is compact,

there is a mn0 E Mp = {mjm is compact and m $ p} such that

xm0y 5 xc. Thus m0y c since x is cancellative, and so

b 5 c. Similarly, c b, so x is cancellative, and thus

invertible. Now, in L p is the unique maximal, and L is

totally ordered. Also, r < p and r is prime, with no primes

between r and p. Let a be an invertible element in L with

r < a 5 p. Then a is invertible, and r < a 5 p < 1. Since
..A A 2 A A A 2 ^
a is invertible and a 1, (a) a, so r < (a) < p. Clear-
2* A5 AS A. AAs A^ 2
ly, rad((a) ) = p. If x and y are compact with xy < (a) ,

then suppose (y) 4 (a) for any n E Z Since L is a
p
v-lattice, we must have y p = rad((a)2), by Theorem 5.2,
A A A, A
part (5). But since y is compact and y 0, y is invertible
A A A2 A
by part (1) of Theorem 5.2, so if y = p, then (y) < y, and
00 A n
A (y) P= p0 is prime with P0 < p. Thus, p0 o r, so for some
n=l
(y)< r < (a), contradicting our choice of y.n
n E Z I (y) < r < (a) contradicting our choice of y.








A ^ 2 ^ 22 "
Therefore, y > p, so y = 1, and x (a) Hence (a) is
"2 2*
p-primary in L p, with (a) d p, and by Theorem 3.19, (a )
2 *
is p-primary with (a ) < p.D



5.6. Definition: A v-lattice L is said to be a discrete

v-lattice if each primary element is a power of its radical.

This definition agrees with the corresponding definition in

commutative ring theory (Gilmer [3]), and as in that case, a

result of Theorem 5.4, part (2), is that a v-lattice L is

discrete if and only if no branched prime p is idempotent
2
(that is, p < p for all branched primes).



5.7. Proposition: Let L be a v-lattice with unique

maximal p > 0. The following conditions are equivalent:

(1) L has ACC.

(2) Every element a with 0 < a < 1 is a power of p.

(3) L is discrete, and p is the only non-zero prime.



Proof: (1) = (2): Let a E L with 0 < a < 1. Since L

has ACC, p is compact, and by Theorem 5.2, part (1), p is

invertible. Since a p, a = p[a:p]. Let a = [a:p3.

Recursively, let an+l = [a n:p] for each n E Z. We have the

ascending chain


a = a0 < a 1 a2 : a ... .

By the ACC, there is a least integer n such that a = ak for

all integers k n. Thus an = a n+ = a n:p]. If an p
= n lpn








then an = pan, and since a -> a > 0, a is compact and

therefore invertible. Thus, p = 1, which is impossible.

Therefore, an $ p, so an = 1, and we have a = pa = p2a2 =
na n
p = p n = p .

(2) => (1): This is clear.

(2) => (3): This is clear.

(3) => (2): In the proof of Theorem 5.5, when we proved

that condition (5) implies condition (1), we actually showed

that in a v-lattice with unique maximal p, any invertible

element with radical equal to p is p-primary. Now, every

invertible element a with 0 < a < 1 has rad(a) = p, since

p is the only nonzero prime. Therefore, all invertible

elements below 1 are p-primary, and since L is discrete, they

are all powers of p. Now, if b is any element p of L with

0 < b < 1, then b is the join of a set of invertible elements,

since L is a v-lattice. Thus, b is the join of a set of

powers of p, so if n is the least positive integer that

occurs as an exponent in this set of powers of p, then

b = pn. D



We now turn our attention to the analogue of Priifer

domains. These are integral domains with identity in which

every nonzero finitely generated ideal is invertible, so we

consider mi-lattices with 1 compact in which each nonzero

compact element is invertible. We will proceed toward the

many arithmetic relations that characterize Priifer domains

in commutative ring theory, but first we consider localiza-

tions in such a mi-lattice.








5.8. Theorem: Let L be a domain lattice. Let

{Pili E I} be the set of all maximals in L. Then the fol-

lowing are equivalent:

(1) Lp is a v-lattice for each prime p in L.

(2) Lpi is a v-lattice, for each i E I.

(3) Every nonzero compact element is invertible.

(4) If a and b are invertible, then a v b is invertible.



2
Proof: (1) => (2): 1 = 1, so each pi is prime.

(2) = (3): Let x be compact, x > 0. Then if (x)i is

the class of x in the localization at pi, (x)i is compact in
A
Lpi, and by Theorem 5.2, part (1), (x)i is invertible in Lpi.

Now, let y be invertible, with y < x in L. Since x is corn-
A A A
pact, if z = [y:x], then (z)i = [(y)i:(x)i], and so
A\ A As
(x) i(z) = (y) i, for each i E I. Thus, if at represents the
i1
maximum element of (a) for each a E L, then y = (xz) for

each i E I. By Theorem 3.22, y = xz, and since y is invert-

ible, x is invertible by Proposition 4.4.

(3) => (4): If a and b are invertible, then since 1 is

compact, a and b must be compact by Proposition 4.9. Thus

a v b is compact, and a v b : a e 0, so a v b t 0. Hence

a v b is invertible.

(4) => (1): Let p be prime in L. In Lp, 1 is compact,

and in proving (5) => (1) in Theorem 5.5, we showed that if

L is any mi-lattice in which 1 is compact, then if x is

invertible, we must have x invertible in Lp for any prime p

of L. Thus each nonzero element of Lp is the join of a set









of invertible elements, and so we need only show that the set

of invertible elements of Lp is totally ordered. Let a and

b be invertible in Lp, and assume without loss of generality

that a and b are compact in L. Then a v b is compact in L,
n
so a v b = v Ck, where each ck is invertible in L. An
k=l k
A A
induction argument shows that a v b is invertible, so a v b
A A A A A A A A
is invertible. Thus a = (a v b)(x) and b = (a v b)(y), so
A A A A A A A A A A
a v b = (a v b)(x v y), and x v y = 1. But p is the unique

maximal of Lp, so since x v y > p, then x = 1 or y = 1. Thus

a = a v b or b = a v b, so a > b or b a, and Lp is totally

ordered.D



5.9. Proposition: If L is a mi-lattice in which 1 is

compact and every nonzero compact element is invertible,

then L is distributive.



Proof: Let a,b, and c E L. For each maximal p, Lp is

a v-lattice by Theorem 5.8, so Lp is totally ordered, and

thus distributive. If d1 = a A (b v c) and d2 = (a A b) v

(a A c), then by Propositions 3.10 and 3.11, for each maximal
^ A *
p of L, d1 = d2 in Lp. Thus, d1 = d2 relative to each

maximal p of L, and by Theorem 3.22, d1 = d2.D



Remark: In commutative ring theory, this is a character-

ization of Priifer domains; i.e., an integral domain with

identity is a Priifer domain if and only if its lattice of








ideals is distributive (Gilmer [3]). In our context, we

must leave it as an open question whether the converse of

Proposition 5.9 is true.

The method of proof used in Proposition 5.9 will be

used in the next theorem as well. That is, we assume we are

looking at the localization at a maximal of L, so that L has

a unique maximal, and show the arithmetic relations under

this condition. Again, because lattices ofextensions preserve

all of the arithmetic relations we consider, as well as

compactness and invertibility, we can then generalize.



5.10. Theorem: Let L be a domain lattice. Then the

following conditions on L are equivalent:

(1) Every nonzero compact element of L is invertible.

(2) For all a,b,c E L, a(b A c) = ab A ac.

(3) For all compact elements a,b,c E L, a(b A c) =

ab A ac.

(4) For all a,b E L, (a v b)(a A b) = ab.

(5) For all compact elements a,b E L, (a v b)(a A b) = ab.

(6) If a,b E L and c is compact in L, then

[ (a v b):c] = [a:c] v [b:c].

(7) If a,b,c are compact in L, then

[(a v b):c] = [a:c] v [b:c].

(8) If a,b are compact in L, [a:b] v [b:a] = 1.

(9) If a,b are invertible in L (and so compact, by

Proposition 4.9), [a:b] v [b:a] = 1.








(10) If c E L and a,b are compact in L, then

[c:(a v b)] = [c:a] v [c:b].

(11) If a,b,c are compact in L, then

[c:(a v b)] = [c:a] v [c:b].


Proof: First, (2) => (3) (4) => (5) (6) (7),

(8) => (9), and (10) = (11) easily. We show (1) => (2),

(1) => (4), (1) = (6), (1) = (8), and (1) = (10) by assuming

L is a v-lattice, by Theorem 5.8. Then, assuming that L has

a unique maximal, we show that (3) =>(5) => (1), that

(7) => (9) = (1), and finally that (11) => (9).

(1) => (2): Since L is totally ordered, we assume that

b < c. Then a(b A c) = ab = ab A ac.

(1) => (4): Again, assume a Z b, so that (a v b)(a A b)

ab.

(1) => (6): Assuming a b, [(a v b):c] = [b:c] =

[a:c] v [b:c].

(1) => (8): If a < b, then [b:a] = 1, so [a:b] v [b:a] =1.

(1) => (10): Suppose a < b. Then [c:(a A b)] = [c:a] =

[c:a] v [c:b], since if xb < c, then xa < xb < c, so

Cc:b] 5 [c:a].

Now we may only assume that L has a unique maximal p.

(5) => (): Let a and b be invertible in L. Then a and

b are nonzero compact elements, so a v b is nonzero and

compact. But (a v b)(a A b) = ab, which is invertible, so

by Proposition 4.4, a v b is invertible. By Theorem 5.8,

part (4), every nonzero compact element is invertible.








(3) = (5): Let a and b be compact in L. (a v )(a A b) =

a(a A b) v b(a A b) < ab, and because (3) holds, (a v b)(a A b)=

(a v b)a A (a v b)b -> ab, so (5) is true.

(9) => (): Let p be the unique maximal of L. We will

show that L is a v-lattice. Since 1 is compact and every

nonzero element is the join of a set of invertibles, we

need only show that the set of invertible elements is totally

ordered. Let a and b be invertible in L. [a:b] v [b:a] = 1

by (a), so either [a:b] t p or [b:a] $ p, where p is the

unique maximal of L. Thus [a:b3 = 1 or [b:a] = 1, so

a < b or b < a.

(7) = (9): If a and b are invertible in L, then

1 = [(a v b):(a v b)] = [a:(a v b)] v [b:(a v b)] =

[a:b ] v [b:a] by (7) and Proposition 2.2, part (d).

(11) => (9): If a and b are invertible in L, then

1 = [(a A b):(a A b)] = [(a A b):a] v [(a A b):b] =

[b:a3 v [a:b] by (11) and Proposition 2.2 part (d).A














CHAPTER VI
GENERALIZATION OF DEDEKIND DOMAINS

We now consider mi-lattices in which every nonzero

element is invertible, and 1 is compact.

First, in light of Proposition 4.9, every element of L

is compact, and so L has ACC. Also, 1-a = a for each a E L,

and 0 is prime, and L satisfies all of the conditions of

Theorems 5.6, 5.7 and 5.8.



6.1. Proposition: If every nonzero element of L is

invertible, then every nonzero prime is maximal.



Proof: Suppose 0 < p < q 1. Then q[p:q] = p, so if

p is prime, then because q > p, [p:q] < p < [p:q], and so

p = [p:q]. Thus pq = p, and since p is cancellative q = I.D



6.2. Proposition: In a mi-lattice L, if x is invertible
n
and x = n p., where each p. is prime in L, then this repre-
i=l
sentation is unique up to the order of the primes.


n m
Proof: Let x be invertible, with x = pi = qj,
i=l j=l

where each pi and each qj is prime. We show that m = n,

and after rearranging, pi = qi for each i n. First, by

60








Proposition 4.4, each pi and each qj is invertible. We
m
proceed by induction on n. If n = 1, then p = l q., so
j=1 I
after rearranging, q, : p since p is prime, and p = ql.
m
If m 2 2, then 1 = (p:ql) = n q., which is impossible, since
j=2

each q. < 1. Thus m = 1 and p = q = x. Now, suppose that

if 1 k < n-l, and if r ,r2,...,rk are all primes in L with
k
H r= yinvertible, then this representation of y is unique
k=l
up to the order of the primes. Then if x is invertible and
n m m
x= ilp. = qj, we have p, x = H q., so after rearrang-
= j=l j=1 3
ing if necessary, p, > q, since p, is prime. Then

pI[qI:pl] = ql, since p, is invertible. If q, < pI, then

[ql:Pl 5 ql since q, is prime, so by Proposition 2.2, part

(a), ql = [ql:pl], and plql = ql, so since q, is cancellative,

pl = 1, which is impossible. Thus q, = pl, so
n m
x = (pl) ( II pi) = (pl)( n q.), and since p, is cancellative,
i=2 j=2
n m
nlp. = H q.. By induction, m 1 = n 1, and so m = n,
i=2 j=2 3
and after rearranging, pi = qi for each i 5 n.0



We now give the major result of this chapter, which is

similar to the characterizations of Dedekind domains, with

one slight variation, which we will discuss after the proof.



6.3. Theorem: In a mi-lattice L in which 1 is compact,

the following conditions are equivalent:








(1) If a E L and a > 0, then a is invertible.
2
(2) 1 = 1 and all nonzero primes are invertible.

(3) Every nonzero element exceeds an invertible element,

and if 0 < a < 1, then a can be uniquely written

as a finite product of maximals.

(4) L is a domain, each nonzero element of L lies

below only a finite number of maximals of L, and

for each maximal of L, L is a v-lattice with ACC
P
(and so L is discrete).
P
(5) L is a domain with ACC, and for each maximal p of

L, L is a v-lattice.
P


Proof: (1) => (2): Obvious.

(2) => (I): Let S = {a E Lia is not invertible}. S

since 0 E S, and S is partially ordered with the order from

L. Let C be a nonempty totally ordered subset of S, with

C = {cili E I}. Let c = v c., and suppose that c 4 S. Then
iEI

c is invertible, and by Proposition 4.9, since 1 is compact,
n
c is compact. Thus c = v c. and since C is totally ordered,
k=l k
c = c0 E C, which is impossible. Hence c E S, so by Zorn's

Lemma, S has a maximal element p. Let a and b be in L with

ab p but a $ p. Then a v p > p, so a v p 4 S, and a v p

is invertible. Thus (a v p)d = p, where d = [p:a v p]. Since

p is not invertible, but a v p is, d must not be invertible,

so d E S. But p 5 d, so since p is maximal in S, p = d. Thus

p(a v p) = p, so (a v p)(b v p) = ((a v p)b) v ((a v p)p) =

ab v bp v p = p since ab p. Again, b v p cannot be






63

invertible since a v p is invertible but p is not, so

b v p E S, and so b v p = p. Thus b p, and p is prime.

But p is not invertible, so p = 0, and every nonzero element

of L is invertible.

(1) => (3): Clearly, every nonzero element exceeds an
2
invertible element, in particular, itself. Since 1 = 1,

maximals are prime, so that in light of Propositions 6.1 and

6.2, we only have to show that each element a with 0 < a < 1

can be written as a finite product of primes. Let S = {a E LI

0 < a < 1 and a cannot be written as a finite product of

primes}. If S p, then since L has ACC by Proposition 4.9,

S must have a maximal element b. Since b E S, b is not

prime, and so b is not maximal in L. Let q E L with b < q < 1.
n
Then q 4 S, so q = l p., where each pi is prime. Also, q e 0,
i=l 1
so q is invertible, and b = q[b:q]. Thus Cb:q] cannot be

written as a finite product of primes, or else b could be.

Then [b:q] E S, and b [b:q], so b = [b:q]. Therefore b = bq,

and since b > 0, b is cancellative, so that q = 1. This

contradicts the choice of q, so S is empty.

(3) => (i): First, by Proposition 4.9, 1 is invertible,
2
and so 1 = 1. Now, let p be a maximal in L. If p = 0, we

are done trivially, so suppose p > 0. Then p is prime, and p
n
exceeds an invertible element a. Since a 0, a = II q., where
i=l 1
n
each q. is a maximal of L. Since IT q. = a p, there must
i=l 1
be an integer i0 < n with qi0 < p. But qi0 is a maximal and
0 0'-








p < 1, so q. = p. Thus p is a factor of a, so p is invert-
10

ible since a is. Therefore each maximal of L is invertible,

so that each nonzero element of L is a product of invertible

elements, and so is invertible.

(1) => (4): By Theorem 5.8, since each nonzero compact

element of L is invertible, then for each maximal p e L, L
P
is a v-lattice. Since 1 is compact in L, each element of L

is compact by Proposition 4.9, so L has ACC, and by Lemma

3.11, for each maximal p E L, L has ACC as well. Let a E L
P
with a > 0. If a = 1, then the set of maximals above a is

empty, and is therefore finite. If a < 1, then by the
n
equivalence of (1) and (3) of this theorem, a = II p., where
i=l 1
each pi is a maximal of L, and this representation as a

product of maximals is unique. But if p is any maximal of L

with a : p, then a = p[a:p] since p is invertible. Thus if

[a:p] = 1, then p = a and p is the only maximal exceeding a,
m
while if [a:p] < 1, then [a:p] a > 0, so [a:pl = n q.,
j=l
m
where each qj is a maximal of L, and a = p( H q.). By the
j=l
uniqueness of the representation, p = pi for some i 5 n, and

so the only maximals above a are those in {pl,...,pn}.

(4) => (5): We only have to show that every element of

L is compact. Let a E L with 0 < a < 1, and let {plP2,...,pn}

be the set of maximals exceeding a. For each positive integer
A
i n and for each b E L, let (b)i be the equivalence class

in Lpi containing b. For each i n, (a)i is compact since
Pi_







A A
L has ACC. If a = v{b IX E A}, then (a). = v{(b).iX E A},
p. x 1 Xi

and for each i : n, there is an integer m. such that
A mi
(a). = v (bX ).. Thus, if B = {bl |for some i < n and for
k=l k
some k i mi, b, = bX k}, then B is finite and for each i : n,
A A
^\ ^
(a). = v{(bXl|IbXE B}. By Theorem 3.22, a = v{b?|b, E B}, and

a is compact.

(5) => (i): By the fact that L has ACC, each nonzero

element of L is compact, and so by Theorem 5.8, each nonzero

element is invertible.0



Recall that in commutative ring theory, an integral

domain with identity is a Dedekind domain if and only if

every proper nonzero ideal is uniquely a product of a finite

number of prime ideals (Hungerford [4]). In the context of

mi-lattices, we must insist that each element a with 0 < a < 1

is uniquely expressible as a product of a finite number of

maximals, as the next two examples illustrate.



6.4. Example: Let P and M be disjoint countably infinite

sets, with P = {pklk is a nonnegative integers, M = {MI.| is

a nonnegative integer}, pk 1 pk2 if kI k k2, and m.1 mk2

if 9i 1 dk2* Let L0 = {0} u (P x M), and order L0 by setting

0 < a for each a E L0, and (Pkl, ) mZ (Pk ,"m ) if k2 < k1 or k1=k2
1 1 2 -2
and Z2 < ZI. L0 is totally ordered, so it is a lattice. For

each family F = {(pk ,m. )|i E I}, v (pk.,m) = (Pk,mz),
i ieIl i







where k = min{k.ili E I} and = min{.ilk. = k}. In particular,
v (pk ,mz) E F, so each element of L0 is compact. Also,
iEI 1 1

0, if {kili E I) is unbounded

(pk+l,m0), if k = max{kili E I} and
A (p ,m ) = {U.ilk. = k} is unbounded
iel i ~i1 1
iEI i i
(pk,m),if k = max{bi.i E I} and
Z = max{P.ik. = k}.

Thus, L0 is an algebraic lattice, with 1 = (p0,m0). Define

a product on L0 by setting 0*a = a.0 = 0 for each a E Lo,
and (p m, m) (Pk ,m2 ) = (Pk+2,m1+ + ). It is easily seen

that this product makes L0 a commutative mi-lattice, with 1
2
compact and l*-a = a for each a E L0. In particular, 1= 1,
so that (p0,ml), the unique maximal of L0, is prime. The
element (Pl,m0) is also prime, for if a > (pl,m0) and
b > (plm0), then a = (POm1 ) and b = (p0,m 2), so that

ab = (P0,m1 + 2) > (PI,m0). Now, if q > (pl,m0), then either

q = 1, which is not prime, or q = (p0,m) for some Z > 1, in

which case q = (p0,ml) and q is not prime unless Z = 1.
If 0 < q < (plm0), then q = (pkmZ), with k > 1 and with
k 1 if k = 1. Thus, q = (pl,m0) (Pklm,), and q is not

prime. Therefore the only primes of L0 are 0, (p0,ml), and

(pl,m0) and each element a of L0 with 0 < a < 1 can be
uniquely expressed as a product of a finite number of primes.

Also, (pl,m0) is multiplicative, for if a < (Pli,m0), then either
a = 0 or a = (pkmI) with k 1, so a = 0(p1,m0) or







a = (Pku,m2Z)(Plm0). Also, (plm0) is multiplicative, since

(Pl'm0) (Pklmk1) e (Pl'm0)(Pk2 ,m2) if kI 1 k2 or Z1 t Z2'

Thus, (pl,m0) is invertible, and so (Pk,mO) is invertible

for each integer k 1. Therefore each nonzero element of

L0 exceeds an invertible element. However, (p0,ml) is not
invertible, since there is no a E L0 with a(p0,mI) = (pl,m),
and so (p0,m1) is not multiplicative. Hence, not every

nonzero element of L0 is invertible.


6.5. Example: It is not even enough to assume that in
L, each nonzero element is a join of invertibles. To see
this, let P, M, and L0 be the same as in Example 6.4, and

let X be a countably infinite set with X n P = X n M =
X n L0 = c, and X = {xnl n is a nonnegative integer} with

xn r x n2 if n1 n2. Let Ln = (L0 \{0}) x {x n}, identifying
00
(Pkm) with (Pkm,xO), and set L = {0} u ( u Ln). For a
n=0
partial order, set 0 < a for each a E L, and (Pk ,m1,xn ) _
~1 1!
(Pk 2,m ,xn ) if n2 < n1 and k2 < kl, or if n2 < n1 and
k2 = kI and n2 + X2 -< nl + k" This is a lattice order on L

which extends the order on L0, and L is an algebraic lattice

with ACC. For a product, set 0*a = a-0 = 0 for each a E L,
and P'km1xn )(Pk2 ,m2,xn = (k+k ,m1 +Z 2,xnl+n2) Then
(k1 m,1 n1 2 2 2 1 2 1 2 1 2
L is a mi-lattice, 1 = (p0,m0,x0), 1 is compact and 1-a = a
for all a E L. Also, (P0,m0,x0) is the unique maximal of L, and
since 12=, (0,mx0) is prime. The only primes in L are
since 1 =1, (pQOm^.XO) is prime. The only primes in L are








0, (p01,ml,x0), (pl,m0,x0), and (p0,m0,xl), and so each

element a with 0 < a < 1 can be uniquely expressed as a

product of a finite number of primes. The elements

(plm0,x0) and (p0,m0,xl) are invertible, and (p0,.nlx0) =
(pl,m0,x0) v (p0,m0,xl), so since the product distributes

over joins, each nonzero element is the join of a set of

invertible elements. Once again, however, there is no

a e L with a(p0,mlx0) = (Plm0ox0), so (P0,ml,x0) is not

invertible. In Figure 6.6, we label (Pk,m,xn) = p m x

with the convention p = m = x = 1 to simplify the

notation, and L0 from Example 6.4 is labeled.







































































k
0 = A p =
k=l


00
n
A Xn
n=l


FIGURE 6.6


/ 2
pmx


- -














CHAPTER VII
MODULARITY

In Chapter II, we assumed modularity for the mi-lattice

L in order to obtain the result that A-prime elements are

primary, which gave us the primary decomposition of Theorem

2.14. However, throughout the rest of this paper, modularity

was not assumed, and it is not clear that modularity was

needed in Theorem 2.14. The question that arises is whether

we can impose other restrictions on L to guarantee modularity.

Of course, the assumption that L is a mi-lattice is not

enough, for any algebraic lattice, and hence any finite

lattice including the five-element nonmodular lattice, is

a mi-lattice with the trivial multiplication a'b = 0 for all

a,b E L. One connection between modularity and invertible

elements is the following result.



7.1. Lemma: If L is a mi-lattice and if a E L is invert-

ible, then L can contain no five-element nonmodular sub-

lattice in which a is the maximum element.



Proof: Suppose L* is a five-element nonmodular sub-

lattice of L, with L* = {a,b,c,d,e}, satisfying a = b v d =

c v d, e = b A d = c A d, and b > c. If a is invertible in

L, then a = 1-a, b = abl, c = acl, d = ad,, and e = ae,








with 1 = bI v dI = c1 v di, e1 = bI A d1 = c1 A dl, and

bI > c1, since multiplication by a is a lattice isomorphism

of L onto [O,a]. Then bl(C1 v dl) = bl(l) = bl, since L

contains an invertible element, but blc1 c1 and

bldI 5 b1 A d1 = el, so that b1c1 v bldI c1 v el = c1 < br

contradicting distributivity of the product.D



This lemma allows us to say that if every nonzero

element of L is invertible, then L is modular. However, we

cannot weaken the amount of invertibility in L beyond this,

for even if L is a mi-lattice in which each nonzero element

is the join of a set of invertible elements, we have the

example at the end of the last section. Here, m = x v p =

x v pm, and px = x A p = x A pm, with pm < p, so L contains

a five-element nonmodular sublattice, and is therefore

not modular.














BIBLIOGRAPHY


[1] Bigard, A., Keimel, K., and Wolfenstein, S., Groupes
et Anneaux R4ticules, Springer-Verlag, Berlin (1977).

[2] Birkhoff, G., Lattice Theory, 3rd Edition, Amer. Math.
Soc. Colloq. Publ., vol. XXV, Providence (1967).

[3] Gilmer, R., Multiplicative Ideal Theory, Marcel Dekker,
Inc., New York (1972).

[4] Hungerford, T. W., Algebra, Holt, Rinehartand Winston,
Inc., New York (1974T).

[5] Keimel, K., "A Unified Theory of Minimal Prime Ideals,"
Acta Math. Acad. Sci. Hung., 23 (1972), pp. 51-69.














BIOGRAPHICAL SKETCH

David Bruce Kenoyer was born on August 27, 1953,in

Seattle, Washington, and is the only son and second of

three children of Howard W. and Beatrice Kenoyer. David

has spent most of his life in Michigan, attending East

Grand Rapids High School before receiving his B.S. from

Central Michigan University and his M.S. from Michigan

State University. While at Michigan State, he met Susan

Tiffany, whom he later married. David enjoys most sports,

especially those which he shares with Sue, such as golf,

tennis, bowling, and softball, as well as camping and

music.








I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.




Joy'ge M-drtinez, Chairman
Associate Professor of
Mathematics
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.



XaeW-J 4^. xA&J-A
David A. Drake
Professor of Mathematics
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.




James E. Keesling J
ofessor of Mathemati&s
I certify that I ha read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.




Neil L. White
Associate Professor of
Mathematics








I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.




Elroy J. Bo duc, Jr.
Professor of Subject Specialization
Teacher Education

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 Council, and was accepted
as partial fulfillment of the requirements for the degree
of Doctor of Philosophy.

August 1982


Dean, Graduate School