Commutative semigroup rings

Chicago Lectures in Mathematics

commutative semigroup rings Robert Gilmer

To the memory of Tom Parker, who kindled my initial interest in commutative semigroup rings.

Chicago Lectures in Mathematics Series Irving Kaplansky, Editor The Theory of Sheaves, by Richard G. Swan (1964) Topics in Ring Theory, by I. N. Herstein (1969) Fields and Rings, by Irving Kaplansky (1969; 2d ed. 1972) Infinite Abelian Group Theory, by Phillip A. Griffith (1970) Topics in Operator Theory, by Richard Beals (1971) Lie Algebras and Locally Compact Groups, by Irving Kaplansky (1971) Several Complex Variables, by Raghavan Narasimhan (1971) Torsion-Free Modules, by Eben Matlis (1973) The Theory of Bernoulli Shifts, by Paul C. Shields (1973) Stable Homotopy and Generalized Homology, by J. F. Adams (1974) Commutative Rings, by Irving Kaplansky (1974) Banach Algebras, by Richard Mosak (1975) Rings with Involution, by I. N. Herstein (1976) Theory of Unitary Group Representation, by George W. Mackey (1976) Infinite-Dimensional Optimization and Convexity, by Ivar Ekeland and Thomas Turnbull (1983)

The University of Chicago Press, Chicago 60637 The University of Chicago Press, Ltd., London ©1984 by The University of Chicago All rights reserved. Published 1984 Printed in the United States of America 93 92 91 90 89 88 87 86 85 84

12 34 5

1980 Mathematics Subject Classification: 13, 20M14 ISBN: 0-226-29391-2 (cloth) 0-226-29392-0 (paper) LCN: 83-51596

CONTENTS

PREFACE........................................................ vii I.

COMMUTATIVE SEMIGROUPS..................................... 1. 2. 3. 4. 5. 6.

II.

SEMIGROUP RINGS AND THEIR DISTINGUISHED ELEMENTS..... 7. 8. 9. 10. 11.

III.

Semigroup Rings.................................... 64 Zero Divisors...................................... 82 Nilpotent Elements................................. 97 Idempotents......................................... 113 Units............................................... 129

Integral Dependence for Monoid Domains........... 147 Monoid Domains as Priifer Domains.................. 162 Monoid Domains as Factorial Domains...............171 Monoid Domains as Krull Domains................... 190 The Divisor Class Group of a Krull Monoid Domain.208

RING-THEORETIC PROPERTIES OF MONOID RINGS............. 225 17. 18. 19. 20.

V.

63

RING-THEORETIC PROPERTIES OF MONOID DOMAINS........... 145 12. 13. 14. 15. 16.

IV.

Basic Notions......................................... 2 Cyclic Semigroups and Numerical Monoids............. 9 Ordered Semigroups................................. 22 Congruences........................................ 28 Noetherian Semigroups............................. 39 Factorization in Commutative Monoids............. 52

Monoid Rings as von Neumann Regular Rings........ 226 Monoid Rings as Priifer Rings...................... 240 Monoid Rings as Arithmetical Rings --- the Case Where S is not Torsion— Free..................... 253 Chain Conditions in Monoid Rings.................. 271

DIMENSION THEORY AND THE ISOMORPHISM PROBLEMS......... 287 21. 22. 23. 24. 25.

Dimension Theory of Monoid Rings.................. 288 R—Automorphisms of R [ S ] .......................... 303 Coefficient Rings in Isomorphic Monoid Rings. I.. 317 Coefficient Rings in Isomorphic Monoid Rings. 11.337 Monoids In Isomorphic Monoid Rings --- a Survey..351

v

1

vi

SELECTED

BIBLIOGRAPHY........................................354

TOPIC INDEX................................................... 362 INDEX OF

MAIN NOTATION.......................................370

INDEX OF

SOME MAIN RESULTS.................................. 37 3

PREFACE

This book contains an account of the current state of knowledge on some topics in the area of commutative semi­ group rings.

Much of the material in the book comes from

literature of the last ten years, and in this regard, a bit of historical information may be in order.

Work on group

rings goes back at least forty years to Higman’s paper [76]. Original work in the area focused on two related problems: the isomorphism problem for groups over a fixed coefficient ring, and the problem of determining the structure of the unit group of a group ring.

In this work, the cofficient

ring was usually taken to be a field (in which case the term group algebra, rather than group ring, was frequently used) or the ring

Z

of integers or (occasionally) the ring of

algebraic integers in a finite algebraic number field --hence a commutative ring in any case.

But restriction to the

case of abelian groups was relatively uncommon, and from a ring— theoretic point of view, most of the work in group rings before 1970 was in noncommutative ring theory. In commutative algebra, semigroup rings arise more natu­ rally than group rings; over

R

for example, each polynomial ring

is a semigroup ring over

R[X^,...,Xn ]

generated over

R

R , subrings of by pure monomials are semi­

group rings, and residue class rings vii

R[{X^}]/I

, where

I

is generated by ’’pure difference binomials”, are also semi­ group rings.

Thus, semigroup rings were seen in the early

1970’s as a general concept under which several classes of important examples in commutative algebra could be considered And with the presentation in [52] of an example of a group algebra

K[G]

that is two-dimensional, non— Noetherian, and

factorial, a new problem area in commutative ring theory was stimulated --- that of determining, for a commutative ring and commutative semigroup semigroup ring of property

E .

S

over

R

S , conditions under which the R

has a given ring— theoretic

In order to solve problems of this type, de­

tailed results about other aspects of commutative semigroup rings --- for example, nilpotents, units, integral closure, and dimension --- were sometimes required.

To a large de­

gree, Chapters II—V recount work done in the problem areas just described. The book is written for a person with a fair background in commutative ring theory, who may know little about the theory of commutative semigroups.

Thus, Chapter I develops

almost from scratch most of the semigroup theory that is sub­ sequently used.

The reader is expected to be familiar with

the material in Chapters 1— 3 of Kaplansky’s book

[83] and

with the material related to primary decomposition in Noetherian rings found in Sections 4— 7 of Chapter IV of [136] The material just cited is a sufficient background in ring theory for Chapters I and II of this book.

In Chapters

III—V, proofs of some results from my book [51] are repeated here, while other results from [51] are merely cited.

Wheth­

er such a result is proved or cited is usually determined

by its importance to the topic in semigroup rings being de­ veloped.

Except for one result from the paper [41] referred

to in Section 20, Chapter V, all the ring theory needed is either developed in this book or it can be found in [83], [136], or [51]. As indicated above, Chapter I is devoted to the develop­ ment of a basic theory of commutative semigroups.

Section 4

of this chapter deals with congruences, and is used through­ out the rest of the book. groups, and is essential

Section 5 treats Noetherian semi­ for the proof in Section 7 that a

monoid ring is Noetherian if and only if the coefficient ring is Noetherian and the monoid in question is finitely gene­ rated. After defining a semigroup ring in Section 7 and p r e ­ senting some basic

properties of such rings, the rest of

Chapter II is devoted to an investigation of zero divisors, nilpotent elements, idempotents, and units of semigroup rings. Known results for polynomial rings serve as a model for the development in Sections 8— 11.

Thus, the symbol

R[X;S]

used in Chapters I and II for the semigroup ring of R , and elements of f =

R[X;S]

S

is over

are written as ’’polynomials”

, with ’’exponents”

s^

in

S .

The general

jaim in §§8— 11 is to determine, under various hypotheses on and

S , equivalent conditions in terms of

in order that of

R [ X ;S]

f

and

should be a zero divisor, nilpotent, etc.

.

Chapters III—V consider problems about semigroup rings that are more global in nature than those discussed in the first two chapters.

In these three chapters,

R[S]

is the

R

X

symbol used to denote the semigroup ring of

S

over

R .

Chapter III treats questions of the form ’’determine necessary and sufficient conditions on the unitary ring monoid

S

in order that the monoid ring

integral domain with property

E" .

R[S]

and the should be an

More specifically, Sec­

tions 12— 15 handle the cases where property erty of being integrally closed,

R

E

is the prop­

a Priifer domain,

domain, or a Krull domain, respectively.

a factorial

In particular, the

example in [52] referred to in the second paragraph above followed resolution of the problem of determining conditions under which

R[S]

is factorial,

as described in Section 14.

Chapter IV has the same basic theme as Chapter III, but prop­ erties

E

that are of interest in rings with zero divisors

are examined in Chapter IV. of being

For example,

E

is the property

(von Neumann) regular in Section 17, while it's the

property of satisfying a specified chain condition in Section

20. Various topics are included in Chapter V. 21,

the (Krull) dimension of

R[S]

, for

S

In Section

a cancellative

monoid, is determined in terms of the dimension of an appro­ priate polynomial ring over group ring

R[Zn ]

R .

The

R— automorphisms of the

are determined in Section 22.

Sections 23

and 24 treat two different cases of the question of whether isomorphism of and

R2 .

R^[S]

and

Specifically,

R 2 [S]

implies isomorphism of

Section 23 provides an affirmative

answer in the case where

R^

and

R2

are fields and

S

tains a periodic element, and Section 24 treats the case where

S

is

Zn

R^

for some

n .

Finally, Section 25 is a

survey of some known results concerning the isomorphism

con­

problem for groups mentioned in the first paragraph of this preface --- that is, does

R[G^] - R [G 2]

imply

6^ = 6 2 ?

The book ends with a selected bibliography, an index of main notation, an index of some main results, and a topic index.

While the bibliography contains 137 items, I did not

attempt to give a complete bibliography.

All but about 25

of the references are cited somewhere in the text, usually in the remarks that conclude each section;

I consulted these

25 items in writing the book, but had no occasion to refer to them specifically.

The index of main results is designed to

help the reader locate them and related results, for I have followed Kaplansky's practice in [83] of calling no result a lemma or a proposition;

each result in [83] is called a the­

orem, but I've included some labeled as corollaries as well. More than any other aspect of [51], I've heard complaints about

(and experienced)

the inadequacy of the topic index;

I

hope that this index is more helpful in tracking items down. Much stimulation for research has come from errors or gaps published in the mathematical literature. four proof readings,

In spite of

I'm sure that Commutative Semigroup

Rings will contain its share of errors, and I can only hope that some of these will stimulate, rather than frustrate, the reader. Most of this book was written during the Fall semester of 1982 while I was on sabbatical leave from Florida State University at the University of Connecticut.

I hereby ex ­

press my appreciation to FSU for its financial support and to UConn for its hospitality and a good atmosphere for work.

In

particular, Gene Spiegel of UConn helped me work out kinks in

the material and in its presentation on a number of occasions I'm grateful for these and for his other deeds of help and kindness.

I finished the book during the summer of 1983,

when I received partial support from National Science Fondation Grant MCS 8122095.

Again I acknowledge with gratitude

this financial support from NSF.

Chapter I COMMUTATIVE SEMIGROUPS

This chapter contains a portion of the basic theory of commutative semigroups.

Other parts of the theory are scat­

tered through the remaining four chapters of the text, closer to the material where applied.

Many ring theorists are u n ­

familiar with the theory of semigroups, and hence the topic, insofar as we subsequently need to use it, is developed in this chapter almost from scratch.

Section 1— 5 are fundamen­

tal for all subsequent chapters, while the material in Section 6 concerning factorization in commutative monoids is not referred to again until Section 14.

2 §1.

Basic Notions

This section introduces basic terminology and notation concerning commutative semigroups that are used in the remainder of the text.

Only two results are stated formally

in the section, and the proof of the second of these is given in outline form. A semigroup is a nonempty set closed under an associa­ tive binary operation. is commutative operation

If

(S,°)

is a semigroup, then

S

(or abelian) if it is commutative under the

° , and

S

has an identity element if there

exists an identity element with respect to with identity is called a monoid.

° ; a semigroup

Except for a brief foray

in Section 23, we deal exclusively with commutative semi­ groups in the text, but comments about the noncommutative case are included in remarks at the end of some sections. The semigroup operation is denoted by strong tradition favoring a different tive) notation.

+

unless there is a

(usually multiplica­

Each commutative semigroup

S

can be

imbedded as a subsemigroup of a commutative monoid.

One way

to accomplish such an imbedding

is to take a symbol

t

in

operation

to the

S

set x e

and extend the semigroup S^ = S u {t} by defining

S^ .

+

on

x + t = t + x = x

S

not

for each

Most of the semigroups considered in the sequel

are assumed to be monoids. If

A

S , then subset

T

and

B

A + B of

are nonempty subsets of the semigroup

is defined to be S

is a subsemigroup of

a semigroup under the operation on T

{a + b|a e A, b e A}

is a subgroup of

S

if

T

S

if

T

.

A

itself is

S , and a subsemigroup

itself is a group.

We remark

3

that if

S

is a monoid,

then a subgroup of

contain the identity element of of subsemigroups of is nonempty,

< B>

with

subset of

of

b^ e B

subset each

I

nS^

If

S

may be empty;

S

s e S .

S , then

B

i .

An ideal of

such that

B + S

S

B ; if

B

S

S

x e I

I * S or

B

^ = 1^

»

is a nonempty

, and

y e I

for

I

is prime if and only if

S , is a subsemigroup of

S , then the radical of (s e s|

S

B c B + S

A

gene-, and

B.

is prime if x,y e S .

An idealIof x + y € I

proper ideal

S\I , the complement of S .

If

I

I

I

is an ideal of

I , denoted

rad(I)

or

there exists

n e Z+

such that

ns e 1 } ; it is easy to show that

S

is a nonempty

generated by

for

in

B

is the ideal of

S

S

defined to be

If

is a monoid, then

of

S .

if

I 2 s + I = {s + i|i e 1}

u(B + S)

the ideal of

is proper if

implies

. Thus,

The intersection of a family of ideals of

S , then

is

nS^

generates a subsemi­

is an ideal, provided it is nonempty.

rated by

S

if

consisting of all finite sums

for each

of

subset of

} is a family

then it is a subsemigroup of

is a nonempty group

S , then

S .

S need not

rad(I)

/I , is

is an ideal of

The first result lists some basic properties of semi­

group ideals. THEOREM 1.1. (1) of

S

Let S

is a

bea

semigroup.

group if and only if

S

is the

onlyideal

S . (2)

If

group of

S

prime ideal (3)

If

I

is an ideal of

such that P I

of

S

S

and

T

is a subsemi­

I n T =

is an ideal of

I

such that

S , then rad(I)

P n T = <J> . is the

4 intersection of the family of prime ideals of I

S

containing

.

Proof. The proof of follows from

(2)

.

(1)

is elementary, and

We prove

suffices to show that

I

(2) .

By Zorn's Lemma, it

is prime if

respect to failure to meet

T .

(3)

I

is maximal with

Thus, assume that I u

{a} u

a,b

e S\I

.

Each of the

ideals

I u

{b} u

(b

+ S)

T , and hence there exist

s 2 ,s 2 e S

such that

quently,

a + s^ e T

a + s 1 + b + s 2 eT

follows that If ment

meets

S

.

b + s2 e T .

I

is prime.

is an additive monoid with identity (zero) ele­ s e

S , then

if there exists

x e

S

s is

such that

said to be invertible s

+ x = 0 ; the set

invertible elements of

S

forms a subgroup

Gis the largest subgroup of

S

containing

sum

S

°f elements of

each if

s^ isinvertible. G * S .

If

H

S\G

is any subgroup of H of

H .

lation

defined

by

a~

s + H

h 6H

S

is

, then

~

cancellative if a, be S .

The

be empty;

if

s

of a semigroup

s + a = s + b set

. A finite

S

containing

C

b

if

C

S

0 ,

a = b + h

s

S

S is said

implies

S

for and

e S .

a = b

to be for

of cancellative elements of

C * <J> , then

of

In fact, if the re ­

is an equivalence relation on

is the equivalence class of An element

and

induces a partition of

s + H

on

S,

is a prime ideal

into disjoint cosets ~

0

of

G

is invertible if and only if

Thus,

then as in the case of groups,

some

Conse­

I n T = 0 , it

Since

a + b £ I , and hence

and if

of

0

and

(a + S)and

is a subsemigroup of

S

may

S

, a

5

a finite sum and hence

I1? ,s. is in i=l 1

S\C

C

if and only

is a prime ideal of

S = C , we say that

S

S

'

if

if each

s.e i

S * C .

C , *

If

is a cancellative semigroup.

A

familiar result from elementary group theory states that a finite cancellative semigroup is a group.

It is clear that

a subsemigroup of a group is cancellative.

Theorem 1.2

proves a form of the converse. THEOREM 1.2. semigroup

If C

S and if

is a subsemigroup of an

each element of C

S , then there exists an imbedding abelian monoid -f(c)

in

T

such that

T for each

(f(s) - f(c)|s e S

(1)

f

and

c e C} .

is cancellative in of

f(c)

c e C , and

additive

S

into an

has an inverse

(2)

T=

The monoid

T

is de­

termined, up to semigroup isomorphism, by properties

(1)

and

T

(2) .

If

S

is cancellative and

S = C , then

is

a group. Outline of a pro of . the familiar one used tegers from the set from let

Zq

property of Denote by and let T

C

To wit,

let

A

of in­

if Sl + c 2 = s 2 + c 1

and

defined by (thecancellative

theequivalence class of

(s,c)

is an abelian monoid with zero element

the mapping

Z

A - S * C

be the set of equivalence classes

the operation

is like

is used in proving transitivity of

[s,c] T

T

of nonnegative integers, starting

be the equivalence relation on

Csi,c i) ~ (s2,c2)

set

in constructing the ring

Peano’s postulates. ~

The construction of

~ ) under

[s,c]

.

* [c,c]

~

The under

[s^,c^] + [s2 ,c2] = [s^ + s 2 ,c^ + c 2 ] , and f : S -- ►

T

defined by

f(s) = [s + c,c]

is

6 an imbedding of [2c,c]

has

element

S

in

inverse

[s,c]

of

T .

If

c e C , then

[c,2c]

in

T

T

,and an arbitrary

can be written as

[s + c,c] + [c,2c] = f(s)

- f(c)

.

It is

determined up to semigroup isomorphism by and

(2) , and if

of

T

f(c) =

clear that T the properties

S = C , then an arbitrary element

has inverse

[c,s]

in

T

so that

is

T

(1)

[s,c]

is a group.

This completes the proof of Theorem 1.1. The monoid

T

constructed as in the proof of Theorem

1.2 is called the quotient monoid of

S

with respect to

By abuse of notation, we write the elements of form

s - c

instead of

be a subset of T

T .

If

f(s) - f(c) S

and

T

in the

we consider

S

to

is cancellative, then the group

of Theorem 1.2 is called the quotient group of

S ; to

within isomorphism, it is the smallest group in which can be imbedded.

C .

S

One way of classifying cancellative semi­

groups is in terms of properties of their quotient groups. For example, abelian groups

arefrequently classified as

being either torsion— free, mixed, or torsion groups. recall the definitions. free if

0

An abelian group

is the only element of

G

is a torsion group if each element of and

G

tive semigroup with quotient group

of

is torsion-

of finite order; G

G

has finite order,

is mixed if it contains elements of infinite order

and nonzero elements of finite order.

that

G

We

G

If

S

is a cancella­

G , then the condition

be torsion— free, for example, can be stated in terms

S ; in fact,

G

is torsion— free if and only if

satisfies the following condition.

S

7 (1.3)

For any positive integer

the equality

nx = ny

implies that

We note that the condition in

n

and any

x,y e S ,

x = y .

(1.3) makes sense whether

is assumed to be cancellative or not; finition of a torsion— free semigroup.

S

it is used as the de­ We will encounter

torsion— free semigroups in Section 3 in determining the class of (totally) ordered semigroups.

We avoid the use of the

term torsion semigroup, preferring the term periodic semi­ group instead.

The definition is given in Section 2 ; we

remark that for a cancellative semigroup is such that

S

S , the definition

is periodic if and only if its quotient

group is a torsion group. Section 1 Remarks The material in Section 1 is both basic and elementary, and hence complete proofs have not been given in the section. Some notation and terminology have been used without ex­ planation.

For example,

inclusion, and where

B

A\B

s

denotes inclusion,

denotes the complement of

is a subset of the set

group isomorphism,

< B

proper in

A ,

A ; the notions of semi­

isomorphic semigroups,

and an imbedding of

one semigroup in another are defined in the expected ways. Many of the notions in Section 1 are meaningful, and have been studied,

for noncommutative semigroups.

But

Theorem 1.2 does not extend to the noncommutative case. Malcev in [94] constructed a noncommutative cancellative semigroup

S

that is not imbeddable in a group; he also

showed that the semigroup ring of

S

over the field

Q

of

rational numbers is a noncommutative ring without zero di-

8

visors that is not imbeddable in a field. Clifford and Preston’s book

[32] is a standard reference

on the theory of (not necessarily commutative)

semigroups.

In addition, Re d e i ’s book [122] treats finitely generated commutative semigroups, and especially the theory of con­ gruences on such semigroups. The

papers

[31],

[39] and [73]

are primarily concerned with the basic theory of commutative semigroups.

9 §2.

Cyclic Semigroups and Numerical Monoids

Two concrete classes of semigroups frequently encoun­ tered in the general theory are the semigroups

<s>

generated by a single element and submonoids of the monoid Zq

of nonnegative integers.

This section contains basic

results concerning semigroups in these two classes. An additive semigroup genic) if there exists {s,2s,...,ns,...} groups.

m

ms

and

s e

S such that

and

infinite order.

S = <s> =

are two cases to consider. ns

Z+

First,

may be distinct for distinct

n ; in this case

additive semigroup

mono­

We determine the structure of such semi­

Clearly there

the elements tegers

.

S is said to be cyclic (or

S

in­

is isomorphic to the

of positive integers, and

The second case, where

S

ms = ns

has

for some

m * n , is covered in Theorem 2.1. THEOREM 2.1. group such that

Assume that ms = ns

for some

smallest integer such that let

S = <s>

is a cyclic semi­

m * n .

ks = rs

Let

for some

k

be the

r < k , and

m = k - r . (1) For

and only

u > v > r ,the equality

ifm

divides

u

us = vs

- v .

(2)

S = {s,2s,...,(k - l)s}has cardinality

(3)

G ={rs,(r + l)s,...,(k

S ; its identity element is between

r

Proof. rs = rs + ms

and (1)

holds if

hs , where

k - l = r + m - l Assume that

- l)s}

k -

1 .

is a subgroup of h

is the integer

that is divisible by

u - v = ma .

, it follows easily that

m .

Since

rs = rs + mas . Hence

vs = rs + (v - r)s = rs + mas + (v - r)s = (v + ma)s = us .

10

For the converse, assume that {r,r + l,...,k - 1}

is a set of

there exist unique integers u

= u1

(modm)

and

proved shows that k

m

u^s

The choice of

k

this set so that .

The part of (1) already-

= us = vs = v^s

,

and the choice

v\

u

= v (modm)

*Therefore

also implies that

{s,2s,...,(k - l)s}

has cardinality

that it is equal to

S .

the mapping

Since

consecutive integers,

u i>v i

ve v^ (modm)

u\ =

implies that

us = vs .

Moreover,

as ----a + (m)

group isomorphism;--since

of

G

Z/(m)

into

(1) shows

also shows that Z/(m)

is a

is a group, then

also a group and its identity is

.

the set

k - 1 , and (1)

of

hs , where

semi­

G

is

h + (m) =

0 + (m) . If

s,r

and

m

are as in Theorem 2.1, then s , and

refer to

as the index and period of the semi­

group

and

m

<s> ; the integer

r + m -1

s , and also the order of the order of <s>

<s>

S

<s> . As

We also

is called the order of in the case of groups,

is the number of elements in

is finite, then

group If

<s>

is the period.

is

called the index of r

m

r

s

<s>

is said to be periodic, and

is periodic if each element of is infinite, then

s

S

If

a semi­

is periodic.

is aperiodic, and

aperiodic if each non-identity element of

.

S

S

is

is aperiodic.

The index and period of a cyclic semigroup determine the cyclic semigroup up to isomorphism. tive integers C(r,m)

r

of index

and r

m and

Moreover,

, there exists

for any posi­

a cyclic semigroup

period m .One such semigroup is

the multiplicative semigroup generated by the class of in

X

Z[X]/(Xr+m - Xr ) ; another is the multiplicative semi-

11

group in q

Z/(2m - 1)Z © Z/qrZ

is a prime greater than

is a groupiff r = 1 of course,

generated by

2m - 1 .

iff C(r,m)

C(r,m)

,

We note that

is amonoid;

the cyclic group of order

hand, each

(2,q)

m .

where

C(r,m)

C(l,m)

is,

On the other

contains a unique idempotent element

(in additive notation, an element

t

is idempotent if

t = t + t = 2t) . The additive semigroup

Z+

infinite cyclic semigroup and

of positive integers is the (Z^,+)

is the monoid ob­

tained by adding an identity element to

Z+

as in Section 1,

On the other hand, the fundamental theorem of arithmetic shows that the multiplicative semigroup

Z+

is the internal

weak direct sum of countably many copies of

(Zq ,+) , in the

sense of the following definition. monoid with zero element submonoids of

0 .

S =

^eA^a^

^

sV ns i=l

equality n 7

E? ..s = E? nt 1=1 a1=1 a. 1 1

S

a

i .

, with

s

ai

for each

S .

finite, we write (Z + ,*) , let let

S^

e S

ai

S

°f

be the submonoid

we

is expressible

in

s

a. 1

i , and if an = t a. 1

S = E S a a

for

and that

a e A , but is not equivalent

S = S,©...©S . 1 n

p^ < p 2 <• • •

S^

need not be

A = {l,2,...,n}

is

To return to the monoid

be the set of prime integers and ^or eac^

Z+

S

(and

for each

implies that ^

In case the set

Unique factorization in

*s a

sum of the family

to these two conditions since elements of invertible in

be an additive

0 , then we say that

The definition implies that r

n l 0 S a = (0) 3*01 3

S

^ a ^aeA

each element of

the form

each

If

S , each containing

is the internal weak direct write

Let

translates precisely to the

* *

12

Statement that

Z+

is the weak direct sum (or product) of

the family

; we return to these notions in Section 6

in the definition of a factorial monoid. We turn to a consideration of submonoids of the additive monoid If

S

.

Such a monoid is called a numerical monoid.

is a nonzero numerical monoid,

common divisor

(g.c.d)

S = dt , where

T

elements of

is

primitive.

T

let

d

0 f the elements of

is isomorphic to

S

be the greatest S .

Then

and the

g.c.d

of

1 ; we call such a numerical monoid

T

To determine all numerical monoids, it suffices

to determine the primitive ones.

Theorem 2.2 foreshadows

what is proved in Theorem 2.4 concerning such monoids. THEOREM 2.2. tive integers.

Let If

x

and

The case where

we assume that 1 < 0 < r^ < a

and

b

a

such that or

a < b . Write

a -1

Therefore

t < q^.

jb - n

-1) <

t > q^

=

and

n

Since

(t

-q^)a

1

is trivial, so

b

, where

is a unit modulo residues modulo

a .

n = ta + r , where

implies that

(a - 1) ,

n = xa + yb .

ib = q^a + r^

is divisible by

jb - (a - 1 )(b

is

3.— 1 r e ^ri^o *

0 < r < a , and hence , for

b

a comPlete set

n > (a - 1) (b - 1) , write

t > q^

be relatively prime posi­

y

0 < i < a -1 .

a , the set If

and

n > (a - 1 ) (b - 1) , then there exist

nonnegative integers Proof.

a

r = rj * t^ien

ta + rj = n < a .

But

a contradiction. +qja

+ r. =jb.

+

Therefore

r^. = (t - q^)a + jb .

Theorem 2.2 implies that a numerical monoid containing two relatively prime integers contains an ideal

K + Zq =

13

{x e ZQ^X “Z 0

^0r some

K e Z+ .

The next result

shows that each nonzero primitive numerical monoid contains a pair of relatively prime integers. THEOREM 2.3. monoid, that

If

S

is a nonzero primitive numerical

S n Z+

There exists a finite subset

with

that

g.c.d. 1

.

1 = x ^ a ^ . . .+x a 11 nn

for

{a,,a0 .... a } l l n

Choose integers .

x^,...,xn

k sufficiently large, each

for each integer & x^ + kxn

and is relatively prime to r

THEOREM 2.4. let

d be the (1)

S

kd S

monoid,

Hence

(a.c.c.)

There exists a finite set

then

Proof.

C

' xn ^an

S . K

such that

S

k > K.

is finitely generated.

such that if

so

e S .

n

g.c.d.of the elements of

for each

k , and 7

be a nonzero numerical monoid and

ascending chain condition (3)

a

There exists a positive integer

contains (2)

Let

such

is positive

+ ^a )a^ = 1 + [k (a i + •••+an _

that

of

Then 1 + [k(a.. + ...+a ,) - x la 1 n -1 nJ n

(x, + ka )a, + ...+(x 1 + ka )a 1 n l n -1 n n -1

S

such

gcd{a,b} = 1 . Proof.

S

a,b e S

then there exist positive integers

ZQ

satisfies the

on submonoids. B

of monoid generators

is any set of generators for

S

as a

B cC . Statement

2.2 and 2.3.

(1)

To prove (2)

of elements of is generated by

S that

are

follows at once from Theorems , we

note thatthe set

less

than

Kd

is finite, and

u (Kd,(K + l)d,...,(2K - 1)d } .

The fact that each submonoid of

Zq

is finitely generated

S

14

implies,

in the usual way,

satisfied in

for

S .

B = {b^} S

(3) , let

S .

Clearly

If .

a.c.c.

on submonoids is

.

To prove ment of

that

{b^}

b^ b^

be the smallest positive ele­ belongs to each generating set

generates

Otherwise,

not in the monoid

let

b^

^

as a monoid,

n

since

Zq

take

B = {b1 ,b2 ,...,bn > .

b^

.

Again it

for any generating set

Continuing this process, we obtain some

then we take

be the smallest element of

generated by

{b^,b2) c c

is clear that

S

satisfies

C

C

of

^ = S

a.c.c.

on

submonoids,

S .

for and we

We have already observed that each nonzero numerical monoid is isomorphic to a primitive one.

The next result

shows that distinct primitive numerical monoids are not isomorphic.

If

and

a homomorphism of S2

S2

are semigroups, then as usual,

into

S2

is a mapping of

into

that preserves the semigroup operation. THEOREM 2.5.

Assume that

numerical monoids.

If

S

and T

h : S -- ► T

are nonzero

is a

homomorphism, then

there exists a nonnegative rational number h(s) = qs

for each

s e S .

or it maps each element of primitive and

h

S

Thus

h

to

0 .

q

so that

is either injective

is an isomorphism of

If S

S

and

onto

T

T

are

, then

S = T . Proof.

Let

s

t = h Cs) , and let it is clear that h ( s s ’) = sh(s')

be a nonzero element of

S , let

q = t/s . Take

s' e S

.

If

s'

* 0 , then

h ( s ’) =

qs '. Thus

h(s') = 0 = = s fh(s)

so

qs1 . that

If

s’ = 0 ,

15 h

is injective or the zero mapping, according as

q = 0 .

and

T

are primitive and

his surjective,

then primitivity of

S

and the inclusion

qS

that

If

q

S

q * 0

is an integer, and primitivity of

equality

T = qS

show that

S .and

T

implies equality,

If

S

and

B

any

B


k e Z+ .

monoids of

for

In contrast

c Z q imply

T

and

the

Thus, isomorphism of

S

are as in part

then the cardinality of submonoid

q = 1 .

and (3)

T

primitive.

of Theorem 2.4,

is called the rank of - 1>°of with

Zq

has

S .

rank

self, need not be finitely generated. (l,k) t <(1,0),(1,1),...,(l,k - 1)>°

k

Zq

If

S

For example, for any

k . to

is a submonoid of

Z

then

S

containing is a subgroup

Z . Proof.

and

Zq

group of integers.

both positive and negative integers, of

for

with it­

It is an easy step to pass from submonoids of

THEOREM 2.6.

The

Z q , we note thatsub­

ZQ© ZQ , the external direct sum of

submonoids of the additive

or

d2 and

Let

= S n Zq

and

are the greatest common S2

S 2 = S n (-Zq)

•

If

d^

divisors of the elements of

, respectively, then since

ZQ

and

-Zq

are

isomorphic, Theorem 2.4 implies that there exist positive integers and

and e ^2

We show that There exist and since integer

K2

such that

^0r eac^

kd^ e

m > K2 .

S = dZ , the inclusion integers

x

and

Let

for each

k ^

d = g .c .d .{d^,d 2 } .

S g dZ

y such that

being clear. d = xd^ + y d 2

xd^ + y d 2 = (x + rd2)d^ + (y - rd^)d2

for any

r , we can assume without loss of generality that

,

16 x > Kx

and

-y >

shows that

.

-d e S

Thus

d e S

as well.

and a similar argument

Therefore

dZ c

S

Theorem 2.6 can be extended to submonoids group

Q .

Before stating this result

and

(Theorem 2.9), however,

material will be referred to again in Chapter 3. x -- ► tx

zero element

is an automorphism of

t e Q .

Q

Recall that a group

H

H

lowest terms.

If

Q by

generated by 1/m .

group of

for each

G

Q

inclusion

H

E

has p ro­

is cyclic. be a finite subset of i

and each

i-s i-n

{1} u {a^/b^}^ Q

is cyclic and is generated

is locally cyclic.

In this proof, we use

(Y) to

generated by the subset

Y

denote the sub­

of

(1,a^/b^,...,an/bn ) c (1/m) is

Q .

n

write

y

1 = a^x + b^y

for integers

l/b1 = x(a1/b 1) + y e (l,a1/b1) . and write

x

and

The

clear. To prove

the reverse inclusion, we use induction on

gcd{b1 ,b2)

contains

m = lcm{b^,...,bn } , then the subgroup of

In particular,

Proof.

H

Let {a./b.}? , i' i i = l

b^ > 0

Q

is locally cyclic if each

finitely generated subgroup of

Q\ {0 } , where

of

is said to have property

In particular,

THEOREM 2.7.

G

Z ,and in consider­

locally if each finitely generated subgroup of E .

First, the

for each non­

ing such subgroups, we frequently assume that

perty

Q ; this

Thus, each nonzero subgroup

is isomorphic to a subgroup containing

Z .

.

of theadditive

we review some of the basic theory of subgroups of

mapping

S = dZ

.

If

n = 1 , .

Then

Ifn = 2 , let d =

d = b^x + b 2y .Then

1/m = d/b^b2 =

x(l/b2) + y (1/b^) e (1 ,a^/b^,a 2/b 2) , so the equality (1/m) = (ljaj/b^,...,an /b )

holds

forn = 1

or

2 ..

17

Assume the result

for

k + 1 , let

lcm{b^, . . . ,b^}

The cases

m’ = n = k

n = k , and in the case where

and

n = 2

so that

n =

m = lcm{m',t^+^}

.

yield the desired conclusion:

1/m e (l/m’,ak+1/bk+1) £ (l.aj/bj,...,ak+1/bk+1) . An arbitrary finitely generated subgroup contained in

({1}

u Y) .

Since

and since a subgroup of a cyclic that

(Y)

is cyclic and

Q

(Y)

of

the latter group is

cyclic

is locally cyclic.

Assume that

(1)

union of an ascending sequence of

is the

is

group is cyclic, it follows

COROLLARY 2.8. H

Q

H

is a subgroup of

Q . cyclic

groups. (2) set

If H contains

{1/pk | p is prime, Proof.

ments of

(1):

H .

Let

If

Statement

k >

H

(2)

the

1/p^ e H} .

an enumeration of the

c..., and

for each H =

ele­

i , then each .

follows immediately from Theorem 2.7. If

both positive and

is generated by

0 , and

{ h ^ b e

c

THEOREM 2.9.

S

is a submonoid of

negative rationals, then

Q containing S is a subgroup

Q . Proof.

zero

then

= (h^,...,h^)

is cyclic,

of

Z ,

It suffices to show that

s e S . Choose

subgroup

G

of

that

g

then

mn < 0 .

Q

t e S

such that

generated by

is a generator of

-s e S

G .

{s,t} If

group (we're using the fact that

G

st < 0 .

no n­

The

is cyclic; assume

s = mg

Thus, the subsemigroup

for each

<s,t>

and of

t = ng , G

is a

admits only two total

18 orders —

one in which

— g > 0 .)

g > 0

and a second in which

-s e <s,t> c S .

It follows that

Because of its connection with numerical semigroups, we conclude this section with a theorem concerning the multipli­ cative semigroup of a commutative ring. semigroups, noted

If

A

and

B

then the external direct sum of

A

and

B , de­

A © B , is the set

A x B , with operation defined by

the coordinatewise operations on THEOREM 2.10. (Z + , 0

and

Let

S

(Z+ ,+ ) .

A

and

B .

be the direct sum ofthe semigroups

If

his a homomorphism

a finitely generated commutative semigroup exist

m,n,a e Z+

Proof.

are

with

m * n

Assume that

into

T , then there

such that

T =

of S

h(m,a) = h(n,a) .

.

For each of the lr

first For

k + 2

primes

p^ ,

1 < j < k + 2 , let

we write

v.

be the

h(pj,l) = E ^ a ^ t ^

•

(k + 1)— tuple

k+ 2 a . Thus, the set {v.}._, is linearly deJ1 JK J J pendent over the rational field Q . An equation of linear (1, a

depencence of EbjVj and

{v^. }

= ^cjvj bj

* c^

=

Eb.v. 3

3

> where b^ for some

implies that n =

over

Qcan be written in and

j .

Z b . = Ec. = a . •Then

c^.

for some

3

integers

s^jvj

k+2 k-; m = II. np.

hfm.a) = I ^ J h f p ^ ' . b ^

= Ec.v. = h(n,a) ; moreover, 3

are positive

The equality Let

the form

= ^cjvj and let

= E*+2b jh(pj ,1) =

m * n

since

b. * c. 33

’’’

j .

THEOREM 2.11. commutative ring

If the multiplicative semigroup of the R

is finitely generated,

then

R

is

19

finite. Proof. acteristic

Consider first the case where p .

In this case we show

finite for each

x

The ring

vector space over

, the field with

assumption that

R

prime char­

that

is (R,#)

is

can be considered as a

linearly

leads to a contradiction, R

has

e R ;this is sufficient since

finitely generated.

of

R

for in that

p

elements.

independent

The over

case the subspace

V

spanned by

{x1 }” is isomorphic as a ring to the ring

XZp[X]

, where

is an indeterminate over Z^ .

{P^}?= ^

is a complete set of nonassociate prime elements of ei e-\r distinct from X , then the mapping (Pj ••-Pv ,a) -- ►

Zp [X]

x [P-^(x)]

ei

X

e ...[P Cx)] v

Theorem 2.10) in {x1 }”

(R,*)

is an imbedding of

x,x^,...,xU ^ .

spanned by {x^}^ * , so finite.

(as in

, contrary to that result.

is linearly dependent, so some

bination of

S

But if

V

xu

Thus

is a linear com­

It follows easily that

is finite and

{x1 }*

is

is also

This completes the proof in the case where

prime characteristic.

V

R

has

The rest of the proof amounts to a

reduction to that case. In the general case, let generators for

(R,*)

x € R , the mapping into

and let

be as above.

For any

is a homomorphism of

S

(R,#) , and hence Theorem 2.10 shows that for some and

has finite additive order. and

S

(m,a) -- ► mxa

m,n,a e Z + , mxa = nxa

K

a finite set of

b

so that

the ideal of

R

Kx^ = 0

m * n .

Thus, a power of

x

We choose the positive integers for

1 < i < k .

consisting of all elements

Let

r e R

I

be

such that

20 Kr = 0 .

The residue class ring hi

finitely many monomials that

h^ < b

for each

i.

R/I hk

is finite since only

in

x^,...,x^

Therefore

R/I

characteristic, and the definition of

I

are such

has nonzero

implies that

R

el ev also has nonzero characteristic c . Let c = p^ ...pv be ®i the prime factorization of c . If R^ = {r e R| p .Jr = 0} for

1 < j < v , then

R,,R~,...,R , and 1 2 v As

Rj

R

R.

j

is the direct sum of the ideals e. has characteristic p .-1 for each vj

is a homomorphic image of

semigroup is finitely generated. finite,

j . J

R , its multiplicative Thus,

it suffices to prove that each

to prove that R^

R

is finite.

is We

therefore assume without loss of generality that the charac­ teristic of

R

is a prime power

pe .

Then

R > pR >...> p e *R > p eR = (0) , and the additive group of p*R/p*+ ^R

is a homomorphic image of that of

i .

if

Thus,

R/pR

R/pR

is finite, then

is a ring of characteristic

image of R/pR

R .

R

p

R/pR

is finite.

for each Now

which is a homomorphic

By the first case of the theorem considered,

is finite and this completes the proof of the theorem. Section 2 Remarks If

B

is a subset of a monoid

of the submonoid of

S

S , then the definition

generated by

in the proof of Theorem 2.4.

B

appears implicitly

The term is defined, of coiirse,

as the intersection of the family of submonoids of taining

B u {0} .

It is denoted by

to adjoining the zero element of S

generated by

S

^

S

con­

, for it amounts

to the subsemigroup of

B .

Additional material on numerical semigroups can be found

21 in [112, Sect.

38],

[75], and [71].

In particular,

there has

been some work on determining the smallest positive integer K

such that

K + Zq

merical monoid b , the value

S .

is contained in a given primitive n u ­ For

S

of rank 2 generated by

K = (a - 1 ) (b - 1)

not, in general, be improved.

a

and

given in Theorem 2.2 can­

Much of the interest in n u ­

merical monoids in commutative algebra stems from the study of subrings of the polynomial or power series ring in an in­ determinate {t

ni k

t

over

R

that are generated over

°f pure monomials.

exponents come from the monoid

of

the ring

^

over

R[t

nl

R .

by a set

The ring in question is just

the set of all polynomials or power series over

polynomials,

R

^

,...,t

nk

]

R

whose

; in the case of

is the semigroup ring

Two recent papers dealing with

rings of this kind are [22] and [49]; polynomial rings are treated in [22], while [49] considers power series. Theorem 2.8 was proved by Isbell, and our proof is merely an elaboration of the proof given in [79].

The question of

whether Theorem 2.8 extends to noncommutative rings appar­ ently remains open.

22

§3.

Ordered Semigroups

An examination of the proofs in Section 2 shows that the order relation on the results.

Z

plays a significant role in several of

In this section we determine the class of

abelian semigroups that admit a relation of total order com­ patible with the semigroup operation.

First we need some

definitions. A relation

~

on a semigroup

(S,+)

is said to be

compatible with the

semigroup operation if

a + x ~ b

+x

a,b,x e S .

trary set

X

transitive,

for

A relation

and asymmetric

(~

is asymmetric if

a = b ) , and a partial order

total

order on X if for distinct elements

either

a ~ b

mean

~ on

implies an arbi­

is called a partial order if it is reflexive,

b ~ a imply that

noted by

a ~b

or b ~ a .

b

and

~

is

a

e

X ,

A partial order is frequently de ­

< and the notation

b < a and

a,b

a ~ b

* a .

a > b

or

b

< a is

used to

To say that a semigroup

S

is

partially ordered (respectively, totally ordered) under a relation

<

means

total order) on

S

group operation on If

S

that < andthat

S —

then it is easy to show that

, so

S

is torsion-

To see this, take distinct elements

say a < b .

Then

a + x < b + x

S is cancellative.

and by induction, S

is compatible with the semi­

S .

free and cancellative.

x e S

<

admits a total order compatible with the semi­

group operation,

a,b e

is a partial order (respectively,

na
is also tors ion-free.

Moreover,

for each

for each 2a < a

+ b < 2b

n e Z+ ,implying that

We prove presently that conversely,

each torsion-free cancellative semigroup can be totally

23 ordered.

The next result shows that to do so, it suffices

to consider the case of a torsion-free group. THEOREM 3.1.

Let

S

be a torsion-free cancellative semi-

group with quotient group

G .

Then

S

admits a total order

compatible with its operation if and only if

G

has this

property. Proof. lation

If

<

G

is totally ordered under

induces a total order on

S

compatible with the

if

S

is totally ordered

^ , then we define a relation

~

on

semigroup operation. under

< , then the re ­

Each element of

G

some

For

s,t e S .

define

g^ ~ g 2

Conversely,

G

is expressible in the form g^^ =

- t^

to mean that

relation

<

s^ + t 2 ^ s 2 +

and that

group operation on G

on S ; we only

and

§2

for

. G

G *

Then

~

that is con­

and extends the

verify that

itagrees with the relation

gl = si - ti = s i - tj

s - t

= s2 ~ X 2

and

is a well defined relation of total order on sistent with the

as follows.

~ is welldefined

^

on S .

= s 2~ t 2

Thus,

if

=s 2 ”t2 ’w^ ere

s^ + t 2 < s 2 + t^ , then

S1

+ t 2 + (s2 + tl) S s2 + tl +

S1

+ t l + s2 + l 2 S1 + t 2

This proves that in

G

we have

is s ~

~

s S2 + tl + S s2 +

is well defined.

s -► (s + c) - c ,where t

if and only

hence if and only if

if

s < t .

(S2 + tl) s2 + t 'l

Z1

■

The imbedding of c e S . For

(s + c) + c <

s,t

(%t + c)

S e S ,

+ c

,

24 Theorem 3.1 leaves us with the problem of proving that a torsion-free abelian group can be totally ordered. more can be proved —

namely, that any partial order on a

torsion-free abelian group order.

G

can be extended to a total

Some additional terminology facilitates the proof.

Thus, assume that

<

P = (x e G|x > 0} . of

A bit

< , and

P

is a partial order on Then

P

G

and define

is called the positive cone

satisfied the following conditions

(i)

and

(ii): P + P c p ; that is,

(i)

(ii) P n (-P) = Moreover, if

<

P

is a subsemigroup of

G .

{0}.

is a total order on

G ,

then condition

(iii) is satisfied. (iii) P u (-P) = G . A subset

L

of

G satisfying (i) and

positive subset order

on

Such a

is called

~

is defined by

a total order if and only if

subset induces a partial

g ~ h L

if

g - h e L ,

satisfies

correspondences between partial orders on subsets of

a

Gthat is compatible with the group operation.

The relation is

of G .

(ii)

G

G

(iii).

and

~

These

and positive

are inverses of each other, and their respec­

tive restrictions to the total orders on

G

and the set of

positive subsets of G satisfying (iii) are also inverses. •v The set of positive subsets of G is partially ordered under

c , and

condition that L2

the inclusion

s l2

~ 2 , the partial order on

, is an extension of the partial order . Theorem 3.2 is THE0RB1 3.2.

is equivalent to the

If

G

induced by induced

by

stated in this vein. Pq

is a positive subset of a torsion-

25 free abelian group

G , then

positive subset of

G

Proof.

positive.

condition g e G

uP

is also

a

Pq

G

(iii) —

that is,

n^ e Z + ;

In Case That

P*

(2)

G - P u (-P) .

ng(-g)^e P

n € Z + , neither 1, let

satisfies

ng

P satisfies

Thus, take

(i)

n^x e P n (-P) = {0} . free.

Thus

plies that

P*

.

satisfies

g € P = P* .

(ii)

-ng

Take Then

e P

Therefore

n^ e Z+ ; is in

P .

= {p+ kg|p e P,k

is clear.

, and since n^g

n^g e P

(1)

for some nor

P* = P + ^

x = ip1 + k xg = -(p2 + k 2g)

-(ngPg + k 2ngg)

is induc­

is contained in a maximal positive

To complete the proof, we show that

for all

say

G , then it is clear that

and consider the following three cases:

for some (3)

is a linearly ordered family of posi­

c , and

P .

P , a

(iii) .

Hence the set of positive subsets of

tive under subset

can be imbedded in

satisfying condition

If

tive subsets of

Pq

x e P* n (-P*)

=0

, and

.

,

nQx = n Qp 1 + k 1 (nQg) =

,it follows

x

e ZQ }

since

that G

is torsion-

maximality

of P

It follows from Case 1 that

im­

-g e P

in Case 2. In Case

3 we again let

-(p2 + k 2g) e P* n (-P*) (ki + k 2)(-g) e P .

in this

By assumption,

k^ = k 2 = 0 , and hence

If

'

case, then

g e P

If x = p^ + kjg p^ + p 2

k^ + k 2 = 0

=

so that

It follows that

(which implies that Case

This completes the proof.

is a family of groups, each isomorphic to

Z , and if family

.

x = p^ = -p2 = 0 .

P* n (-P*) = {0} , so again 3 does not arise).

P* =P + ^

G = E^^Z^

{Z } , then

a

is the complete direct sum of the

P = {{n } I n > 0 a 1 a

for each a)

is a

=

26 positive subset of {a } ^ (b } a a

if

a

dinal order on G

G . a

G .

> b

The induced partial order, defined by for each

a

One extension of the cardinal order on

to a total order is the lexicographic order on

fined by well-ordering the set if a

a , is called the car* ---

a

a > b a0 a0 * b

.

a

A , then setting

for the first element

an 0

The external weak direct sum

family

is a subgroup of

G , de­

of

A

for which

G„ = Ew .Z 0 aeA a

G , and the orders on

of the Gq

in­

duced by the cardinal and lexicographic orders,respectively, are called the cardinal and lexicographic orders on Another total order on

Gq

in frequent use is the reverse

lexicographic order, whose positive cone consists of the nonzero elements of positive.

Z

0

and

whose last nonzero coordinate is

We note that the assumption that each

morphic to above;

Gq

Gq .

plays only a minor

Z^

is iso­

role in the definitions

the definitions are meaningful for any family

of totally ordered semigroups. is isomorphic to

It is the case where each

Z^

Z , however, that is used in the next

corollary.

COROLLARY 3.3.

If

a ^ree subset of a

torsion-free abelian group

G , then there exists a total

order

0 < ga

^

on

Proof. (g 6a }

G

such that

for each

The subgroup

is isomorphic to ^

G

Ew aeAAZ a 9, where

Extending the cardinal order on &

E

a

gene**ated by

Za a Z

.(g ) aeA &a

.

for each

a .

to a total order

yields the desired conclusion. For the sake of future reference, we state formally the

27 result proved concerning orderability of a commutative semi­ group . COROLLARY 3.4.

The abelian semigroup

S

admits a total

order compatible with its semigroup operation if and only if S

is torsion-free and cancellative. Also for the sake of reference, we record one consequence

of the proof of Theorem 3.2. COROLLARY 3.5.

Assume that

the torsion-free group nx k Pq

for each

positive subset and

P

G

and that

n e Z+ . of

G

PQ

Then

is a positive subset of x e G

Pq

such that

is such that

can be extended to a P

satisfies

(iii)

—x e P . Section 3 Remarks The result

totally ordered

that atorsion— free abelian group can be is due toLevi

[92], but the method used in

the proof of Theorem 3.2 is closer to that of [80]; the class of lattice— ordered abelian groups is also discussed in [80]. For

additional information concerning the question of what

nonabelian groups can be totally ordered, see Chapter 3 of [46].

In particular, a (multiplicative) group with the p r o ­

perty that

x * y

implies

x11 * y11

for all

n e Z+

need

not admit a total ordering compatible with the group operation [93].

28

§4. If

S

Congruences

is a semigroup,

the homomorphisms defined on

and the homomorphic images of determining the structure of

S

play an important role in

S .

An equivalent and con­

venient way of considering homomorphisms on the notion of a congruence on congruence on

S

S

S

is through

S , defined as follows.

is an equivalence relation on

compatible with the semigroup operation.

S

A

that is

Theorem 4.1 states

the basic relationships between homomorphisms and congruences. The proof of Theorem 4.1 is routine and will be omitted. THEOREM (1) note by s

4.1.

If

~

Let

S

is a congruence on

eS , and let

(2)

S/p

under

de ­

~ foreach

S/~

is an

[a] + [b]= [a + b]

defined

by f(s) =

S/~ ; moreover,

[s]

, is

f(s^) = f(s2)

si ~ s 2 * h : S -- ► T

T , define the relation

[s] -- ► h(s) s e S

onto

Conversely, if

h(a) = h(b) groups

S

s e S,

under

Then

the operation

f : S -► S/~

a homomorphism of and only if

of s

S/~ = {[s ] | s e S} .

and the mapping

onto

S , then for

[s]the equivalence class

abelian semigroup under

S

be an additive semigroup.

.

Then

and

T

, where

p

is a homomorphism of on

is a congruence

p

S

by

on S

a p b and

if

thesemi­

are isomorphic under the mapping [s]

denotes the equivalence class of

p .

The semigroup

S/~

defined in

called the factor semigroup of

S

(1)

of Theorem 4.1 is

with respect to

Theorem 4.1 shows that, to within isomorphism, semigroups represent all homomorphic images of

~ .

these factor S .

if

29

If

and

are congruences, then we write

~2

if

a

b

implies a ~ 2 b

and

~2

as

subsets of

and

only of

f°r

a >b €

S , the relation

is contained in

form of the relation

< ~2

Thus,

S .

is the assertion that the onto

S

is

the smallest congruence on

gruences on family

S

S

S

is well

are related, and

is the identity congruence

The intersection of any family of con­

is again a congruence on

^~a ^aeA

S/~ 2

S * S , the universal con­

gruence under which any two elements of

.

is a partial

is well defined, it is a homomorphism).

The largest congruence on

i = {(s,s) | s e S}

<

holds if

Another equivalent

[s ]^ ---► [s]2 ofS/~^

defined (if the mapping

Considering

< ~2

~2 .

order on the set of congruences on

natural mapping

S .

< ~2

S ; hence any

congruences has both a least upper bound

(denoted

l.u.b.{~a > or lub{~a)) and a greatest lower bound

(denoted

g.l.b.{~a > or glb{~a >) in the set of congruences on

S .

The relation

means that

a ~

a

~ = glb{~a > b

note that any subset

for p

each of

iseasy to describe: a.To describe

S x S

lub{~

a

} , we

generates a congruence on

S , namely, the intersection of all congruences on contain

p .

lub{~a )

is the congruence generated by U

of lub{~a >

a ~ b

S

that

We proceed to describe this congruence; since

follows.

The congruence

~

, a description generated by

p

is

constructed in three steps. 1. and

i

tains and

Set

Pq = p u p_1 u i , where

is the identity congruences on p

S .

Then

and is both reflexive and symmetric.

pQgenerate the same congruece on 2.

p 1 = { (b ,a)| (a,b) e p}

Set

Pq

con­

Clearly

p

S .

p1 = p0 u {(a + c,b+ c) | (a,b)

e pQ

and c e

S} .

30 Then

is reflexive, symmetric,

semigroup operation on { (a + c,b + c) | (a,b)

S

.

IfS

and is compatible with the is a

monoid, then

p-^ =

e p and c e S} since pn is 0 u Again it is clear that Pq and

tained in this set.

conp^

generate the same congruence. 3.

Let

such that

~ =

((a,b) |

a = an , b 0

= a^, and t

i = 0,1,...,t - 1} . rated by

there exist

aQ,a^,...,a^

fa. ,a. .. ) e p i i+I 1

Then ~

is the

for

congruence on

S

gene­

p . P =U ~ is the union of a family of cona a 7

In case gruences on

S , thenp = pQ =

need be applied to A congruence

p ~

p^ , and only Step3 above

in order to obtain on

S

there exists a finite subset

that

is the

~

Noetherian

S

P

of

congruence generated by

if each congruence on

As usual, each congruence on only if

lub{~a )

satisfies

a.c.c.

S

S

p

S

S

is finite.

such

, andS

is

is finitely generated if and on congruences, which is true < ~2 < * * *

In Section 7 we prove that a

is Noetherian if and only if

rated as a semigroup.

S x S

is finitely generated.

if an only if each strictly increasing set congruences on

.

is said to be finitely gene­

rated if

monoid

€ S

S

is finitely gene­

We return briefly to a consideration

of least upper bounds and greatest lower bounds of congruences in Section 5.

The rest of this section is devoted to an

examination of some examples of congruences that naturally arise in the remainder of the text. THEOREM 4.2.

Assume that

T

is a subsemigroup of the on

semigroup

S .

Define a relation

~

a + t = b

+ t

for some t e T .

Then

S

by

a ~ b

~is a congruence

if

31 on

S

and

If

p

is any congruence on S

S/p

[t]

is cancellative in such

S/~

for each

that the image of

consists of cancellative elements of Proof.

a ~ b

and

for some

Clearly

~

t1 »t2

6 T

>

so

c.

b +

s + t^

If

[a] + [t] = [b] + [t]

then a ~ b

a + ft2 +

a + t^ = b + t^ s e S

so that

for each

[a] = [b] . t e T .

p > ~. If

b + t2 = c + t2

= c + Ct2 + ti^

and

a + s + t^ =

~

congruence on

is a

a,b e s

for some

Thus,

and

in

implies

for some

a + t + t ^ = b + t + t ^ and

S/p , then

a + t^ = b + t^

a ~

for each

T

is reflexive and symmetric.

b ~ c , then

Moreover,

t e T .

[t]

and

t^ e t

t e t

~ < p

,

, and hence

is cancellative in

The assertion that

S .

for

p

S/~

as de­

scribed is clear. In the case where S

~

defined on

as in Theorem 4.2 is called the cancellative congruence on

S . T

T = S , the congruence

For a subsemigroup is cancellative in

respect to semigroup

T T

T

of

S

such that each element of

S , the quotient monoid of

was defined in Section 1. , the quotient monoid of

{[t ] | t e T}

S

defined in Theorem 4.2.

, where

with

For a general sub­ with respect to

is defined to be the quotient monoid of the subsemigroup

S

S/~ ~

T

with respect to

is the congruence

The definition is reminiscent of the

definition of the quotient ring

of a commutative ring

with respect to a multiplicative system

R

N , and indeed there

is a connection between the two, as the next result shows. THEOREM 4.3.

Assume that

in the commutative ring under multiplication,

R .

N

is a multiplicative system

Regarding

R

the quotient monoid of

as a semigroup R

with respect

32 to its subsemigroup the ring

N

is the multiplicative semigroup of

.

Proof. We recall the definition of R . Let I be ----N the ideal of R consisting of all r 6 R such that rn = 0 for some from

R

n e N

and let

onto R/I .

system in
<J>(R)

<j> be the natural homomorphism

Then

and

(N)

is a regular multiplicative

is defined to be N = (<J>0)/<J> (n) | r e R,n e N} . The elements of the

<J>[NJ quotient monoid of in the form

R

R

[r]/[n]

, where

x e R

with repect to

a ~ b

if

an = bn

we show that

for some

yn = xn iff

The next result

N

can be represented

denotes the class of

~

is defined on

n e N .

for each for some

y - x e I

[x] = x + I = <J>(x) ,

[x]

~ , and

[x] = <J>(x)

if and only if (y - x)n = 0

with respect to

by

To complete the proof,

x e R .

Thus,

n e N .

iff

R

But

y e x + I .

y e [x]

yn = xn

iff

Therefore

and this completes the proof. is essentially contained inSection

We restate it here in the language of congruences.

1.

The

proof should be familiar from group theory, and hence it is omitted. THEOREM 4.4. is a subgroup of a ~ b

Assume that S

containing

to mean that

is a congruence on

S

a = b + h

is a monoid and that 0 .

For

for some

a,b e S, define h e H

S , [a] = a + H , and if

of all invertible elements of vertible element of We remark that if

S , then

H

[0]

H

. Then

~

is the group

is the only in­

S/~ . S is a group, then the only congruences

33 on

S

arise as in Theorem 4.4 —

that is, as congruences with

respect

to a subgroup of S .

gruence

on the group

S and

let H

= {s e S | s

H

subgroup of

S ,for

a ~0

,b ~ 0imply that

b so

~ 0 ; also,

is a

a + b ~

a + b

To see this,

Thus

H is asubgroup of

S and

a - b

£H

.

iff a 6 b + H

There is an analogue,

a

let

-a ~ 0 a ~ b

~

be a con­

~ 0} .Then

- a

iff

so

0~

a - b~

- a.

0

iff

for torsion-freeness, of the can­

cellative property in Theorem 4.2.

We can kill more than the

torsion-free bird, however, with a single stone. THEOREM 4.5. and that tegers.

S

is an additive semigroup

is a multiplicative semigroup of positive in­

For

for some m[a]

M

Assume that

a,b e S , define

m e M .

= m[b]

Then

for some

~

a ~ b

to mean that

is a congruence on

[a],[b]

e

S/~

ma = mb

S , and if

and some

m e M , then

[a] = [b] . Proof.

The reflexive and symmetric properties of

are immediate.

If

m^a - m^b

andm^b = m^c

mm a = in m e and m in e M ,so 12 12 12 Moreover, m^a = m^b implies that for

x e S ; hence ~

m[a]

= m[b]

means

nm e M , it follows that In case

M = Z+

torsion-free and such that

S/p

~

m^(a + x) = m^(b + x)

nma = nmb a ~ b

, then

~is alsotransitive.

is a congruence on that

S .

for some

and

The equality ne M .

Since

[a] = [b] .

in Theorem 4.5, the semigroup is the smallest congruence

is torsion-free.

~

p

S/~ on

is S

In considering nilpotent

elements of a semigroup ring over a ring of prime character­ istic

p

in Section 9, we shall encounter the case of

34

Theorem 4.5 where

M = {P 1}i=0

iS th° S6t °f powers of p ;

the congruence is denoted by

~ in

ferred to as

If ~

p— equivalence.

congruence on

S , we say that

S

this case, and is

re­

is the identity is

p— tors ion-free.

Another congruence encountered in Section 9 is that of asymptotic equivalence, defined by setting a ~ b if there + exists K e Z such that ka = kb for each k ^ K . Theorem 4.6

provides an alternate way of viewing this re­

lation, and Theorem 4.7 is the analogue, equivalence, of Theorem 4.5.

for asymptotic

We say that

S

asymptotic torsion if distinct elements of

is free of S

are not

asymptotically equivalent. THEOREM 4.6. na = nb

Assume that

for some

a,b € S

n e Z + . Let

positive integers

n .

additive semigroup

Then

M

M is

are such that

be the set ofall such a subsemigroup of

the

Z+ , and the following conditions are

equivalent. Cl)

a

and

b

are asymptotically equivalent.

(2)

M

is primitive.

(3)

T h e r e exist

n ^ , n 2 , .. . ,n^ e

M

such that

g.c.d. {nj_,. . .n^} = 1 .

(4)

There exist

m,n e M such that

m

and

n

are

relatively prime. Proof. Conditions

That (1)

M and

is closed under addition is immediate. (2)

are equivalent by the definition

of asymptotic equivalence, and the equivalence of and

(4)

(2) , (3) ,

follows from Theorems 2.2 and 2.3 and from the de ­

finition of primitivity.

35 THEOREM 4.7.

Assume that

~

is

the relation of

~

is

a congruence on

asymptotic torsion. that

S/p

S

asymptotic equivalence on S , and the semigroup If

p

is any congruence on

We prove only that

torsion.

Thus, assume that

[a]

[b]

and

there

exists

such that

k 2 €Z+

[b]

in

S/~

m,n

k 2na

= k 2nb .

it follows

[a] = [b]

and

S/A

Since

such

Then there ma ~

mb

and

k^ e Z+

such

By the

same token,

relatively prime to k^m

form (4.6) that S/~

Does there

k2

S

are such that

so that

g.c.d.{k^,n} = 1 . , with

free of

p > ~ .

Theorem 4.6 shows that there exists

k^ma = k^mb and

Then

is free of asymptotic

are asymptotically equivalent.

that

that

S/~

[a],

exist relatively prime integers na ~ nb .

S. S/~ is

is free of asymptotic torsion, then

Proof.

prime,

is a semigroup and that

andk2n

k^m

,

are relatively

a ~ b .

Therefore

is free of asymptotic

torsion.

exist a minimal congruence A

is torsion-free and cancellative?

on

S such

The answer is

affirmative, and the congruence in question is, in this case, the least upper bound of the cancellative congruence on and the minimal torsion-free.

congruence

A

on

Rather than regard

bound, however, gruence

p

S A

it's easier to define

such that as this

S

S/p is least

it directly.

upper

The con­

will reappear in Section 8 , where zero divisors

of semigroup rings are considered. THEOREM 4.8. S

as follows:

such that

na +

Define the relation a A b if there exists x = nb + x .

Then

and is the smallest congruence on

A n

A is S

on the semigroup e Z+ and

x e S

a congruence on

such that

S/A

is

S

36 torsion-free and cancellative. Proof. a A a

since

nb + x and

We verify all the details for a change. a

+ x=a + x

implies

for each

nb +x = na + x .

If

n^a

na

+ x =

+ x^ = n^b + x^

n 2b + x 2 = n 2c + x 2 , then it is easy to check that

n ln 2a + n 2x l + n lx 2

=n ln 2c + n 2x l + n lx 2 *

na + x = nb + ximplies that for each

y e S .

in S/A , then

for some

and

m

Similarly,

and

x . S/p

m

[a] = [b] .

na A nb

and

if

= n(b + y) + x

a Ab

and

means that

[a]

= [b] .

m(a + c) + x =

S/A

is torsion-free and

is torsion-free and can­ x = nb + x

for

Hence n[a]p + [x]p = n t^]p + tx ]p

is also torsion-free,

If

so mna + x = mnb + x

na +

is cancellative,

S .

x ; again this implies that

S/p

a A b , then

Moreover,

is a congruence on

Therefore

Finally,

cellative and if

Since

A

x , implying that

for some

cancellative.

and

+ y) + x

[a] + [c] = [b] + [c]

m(b + c) + x a A b

n(a

Therefore

n[a] = n[b]

n

x e S , and

Thus

some

in

s /p

n[a]p = n[b]p , and since

[a ]p = [b]p .

Therefore

• it

a p b

and

this completes the proof. We consider one other class of congruences on a semi­ group S .

S

before ending this section.

The relation

either

a = b

or

on

S

where

can be thought of as x+°° = «>+x

= «>

x,y e S\I , their sum in

be an ideal of

is a congruence on

called the Rees congruence modulo

S/I

I

defined by setting

a,b e l

is sometimes denoted by

Let

S/I S\I for S/I

I .

a ~ b

if

S ; it is

The factor semigroup

instead of

S/~ . Setwise,

, together with a symbol each

x e S/I

is either

and where,

x + y

or

00 ,

00 for

,

37 depending upon whether

x + y e S\I

factor semigroups

have some of the properties that one

S/I

or

x + y e I .

The

would anticipate of them from ring-theoretic notation. example,

if

For

{Ja }

is

the set of ideals of

then

Ja -- ► Ja/I

is

an inclusion-preserving bisection from

{Ja )

onto the set of ideals of

S/Ja ~ (S/I)/(Ja/I)

for each

S containing

I ,

S/I , and a .

This result can be used

to show that a Noetherian semigroup satisfies the ascending chain condition for ideals, but we give a proof based on first principles in the next section. A straightforward description of where

~

is a Rees

congruence,

lub{p,~}

, in the case

is contained in

the next re­

sult. THEOREM 4.9.Assume that the

semigroup

the ideal

I

there exist Then

S , where of

and

~

are congruences

y

be the relation a p c , c ~ d,

{(a,b) | and

d p b}

and

It is clear that

b y e ,

y

is reflexive, symmetric, and

then there exist

w,x,y,z e S

a

p w, w ~ x., x p b , b p y , y ~ z ,

y

p z , then the transitive property of

nition of and

y

y * z

therefore

imply that so

w ~ x

w ~ z

gruence on y < lub{p,~)

S .

.

.

that it is compatible with the semigroup operation. a y b

on

is the Rees congruence modulo

e S such that

y = lub{p,~} Proof.

~

S . Let

c,d

p

a y e .

and

and again

and

z p c . p

y ~ z imply that

Since the relations

Thus

so that

If w p x

case,

w* x

w,x,y,z £ y

I ;

is a con­

p < y , ~ < y ,

are clear, it follows that

or

and the defi­

In the other

a y e .

If

y = lub{p,~}

and .

38 Section 4 Remarks It is traditional to blur the distinction between con­ sideration of a relation A x A

p

on a set

A

as a subset of

or as a rule by which certain pairs of elements are

related.

Considered as a subset of

A x A , we write

p c A x A ; considered as a rule, we write

a p b .

We have

maintained this dual consideration of congruences in Section n 4 and will continue to do so. If is a finite set of relations on a *?=^ p ^

A , then

glb{p^}^

is frequently denoted by

; considered as a subset of

A x A , we have

a1? p. = n? p . The corresponding lattice-theoretic noi=l l i=l i „ n n tation for lub{p^>1 is vi=iPi • For relations, n n v, p. is u,p. as a subset of A xA ,but whereas the 1 i 1 l ’ finite intersection of congruences on

a semigroup is a con­

gruence, the set-theoretic union of finitely many congruences need not be transitive.

39 §5.

Noetherian Semigroups

Recall that a semigroup

S

is Noetherian if the ascend­

ing chain condition for congruences holds in main purpose

S .

While the

of this section is to prove Theorem 5.10, which

shows that a Noetherian monoid is finitely generated, the material divides rather naturally into three related parts. Some basic consequences of the developed in the first part.

a.c.c.

on congruences are

The second part includes

several new concepts and is devoted to proving Theorem 5.8, a form of Primary Decomposition Theorem for congruences on a Noetherian semigroup.

Theorem 5.10 and its proof constitute

the third part of the section. THEOREM 5.1. (1)

The

each ideal (2)

be a Noetherian semigroup.

for ideals of

S

is satisfied.

Hence

is finitely generated. S

is a monoid and if

vertible elements of G

S

a.c.c.

ofS If

Let

S , then the

G

is the group of in­

a.c.c.

for subgroups of

is satisfied. (3)

PC = S\P (4) then

A

If

P

is a proper prime ideal of

S , then

is Noetherian. If

A

is an ideal of

S

such that

A

is a monoid,

is Noetherian.

Proof. Let I cl <=... be an ascending chain of ----1 “ 2 “ ideals of S . Letbe the Rees congruence modulo 1^ for

each j . Then

that for if x

= ... Ik


x

~i " .

~2 ~

» so there exists

This implies that 1^ = Ik+1 = ...

,then fory eI]c+l^Ik ^ky

’

and

x € Xk * we

» contrary to assumption.

,

have

ksu

40 The proof of c

E ...

by 7

i

shows that

that

is similar to that of

a

b

i

if

a e b + H. . i

is a congruence on

-

if

•••

•

Moreover,

If

~

G ,

Theorem 4.4

S , and it is clear that

an equality

H^. = Hk + -L , and this establishes (3)

(1) :

is an ascending sequence of subgroups of

then define

< ~2

(2)

is a congruence on

= ~^+l implies

(2) .

P

, then it is easy

verify that

~ u(P * P)

assumption that

P° is prime is used only to assert that

is a semigroup).

Moreover,

Thus, if

< ~2 - • • •

gruences on

PC , then

quence on

is a congruence

S .

on

agrees with

to

S (the

~

on P

Pc

Q

is an ascending sequence of con­ < ...

Since

= ~£+1

is an ascending se­

implies that

= ~jc+1 ,

this completes the proof. (4)

Let

gruence

z

on

follows.

For

be the identity element of

A , define the relation s,t e S , s

t

on

means that

We verify that

is

a congruence on S

~ * n ( A * A ) = ~ .

If

s e S ,

(s + z)~(s + z) ; therefore symmetric If

s

t

and if

imply that

[(s + z) +

(x + z)]~[(t +

a,b e A , then n (AxA) a.c.c. of

A .

on

= ~ .

~

s+

For a con­

S

as

(s + z)~(t + z) . and that z e A

and

Similarly,

is symmetric

is

(transitive).

(s + z)~(t + z)

and

s + x+ z =

and a

s .

x e S , then

x + z e A

(s + x)~*(t + x)

then

s

(transitive) because

A .

z) + (x + z)] = t + is a congruence on

b iff (a + z)~(b + z)

x +z . S .

S implies

a.c.c.

If

iff a ~ b .

It thenfollows as in the proof

congruences of

Hence

of

Thus,

(3) that

on congruences

41 With regard to

(2)

of Theorem 5.1, we state the fol­

lowing result concerning Noetherian groups as a corollary. COROLLARY 5.2.

The following conditions are equivalent

in an abelian group

G .

(1)

G

is Noetherian.

(2)

G

satisfies

(3)

G

is finitely generated.

Proof.

a.c.c.

on subgroups.

Since congruences on

G

are in one-to-one order-

preserving correspondence with subgroups of (1)

and

(2)

are

equivalent. Clearly

and it is well known that each

G , conditions

(2)

subgroup

implies

(3) ,

of a finitely gener­

ated abelian group is finitely generated, so

(3)

implies

(2 ) . THEOREM 5.3. a.c.c.

Assume that S

for ideals

or there exist elements

Choose b =

a.

Set

a,y e A

suchthat c

T = < a , , A 17 7 n

b e S\T .

Then clearly

+ s. , where 1 *

b k T , then

monoid satisfying

and that A is an ideal of S generated n c •Then either S =

by the finite set

Proof. -----

is a

s^ k

and assume that

be A .

1 < i, < n 1 T so

>

and

s^ e A

a = a + y

1

e S .

si= a i

Since

+ s2

oo

By induction, there exists

{i.}^ g {l,2,...,n}

and

oo

{s .}1 E A

such that

s- = a^

3

+ s-+^

for each

j .

+ Sss^

+ Sc...

of

j+l

The ascending chain

b + Scs^

principal ideals of

S

stabilizes;

say

T *

Write

s,

and

.

s^. + S =

’

S.

42 Thus,

a = sv+1 K 1 a + y .

and

COROLLARY 5.4.

y = t + axk+l

are in

A

and

a =

A cancellative Noetherian monoid is

finitely generated. Proof.

Let

S

be such

of invertible elements of 5.2 shows that

S

a monoid and let

S .

If

G= S ,

G

be the group

then

Corollary

admits a finite set

generators

as a group, so

{g }n u {-g }n is a finite set of generators i 1 i 1 for S as a semigroup. If G * S , then let A be the c ideal G of S . Theorem 5.1 shows thatA has a finite

set

generators as

an ideal of

S

and

that

G

finitely generated both as a group and as a semigroup. Theorem 5.3, either ments

a,y

a = a + y

of

A

such that

By

or there exist ele-

a = a + y .An

equality

is impossible in this case since the cancellative

property of of

S = 1' * m

is

S

would imply that

y

S , contrary to the fact that

S =

and

S

is the identity element y e A = G

.

Therefore

is finitely generated.

We turn toward a proof of Theorem 5.8, a form of Primary Decomposition Theorem.

The statement and proof of this re­

sult use several new concepts. denote

by U

S

p(s)

{(a,b) |(a

of p(s)

p

S

S

the universal congruence

is a congruence on on

Let

to be

transfer

and

easily to

is a congruence on

s

be a semigroup. S * Son

S .

We

If

p

e S , we define the relation

+ s) p (b+s)} .

Properties

p(s)

so that, in particular, o S .We denote by [p] the set

{s 6 S | p(s) = U} = {s e S |(s

+ a)

p (s + b)

for all o

a,b e S} , and

[p]

is defined to be

{s e S| ns e [p]

for

43 n e Z+ } = {s e S | there exists

some

(ns + a) p (ns + b) [p]° .

such that

a,b e S} , the radical of

Theorem 5.5 states some of the basic properties of

p(s) , [P]°

, and

[p] .

THEOREM 5.5. P , P 1 >...,P (1) If

for all

n e Z+

S

Assume that

S

is a semigroup and that

are congruences on

If

P * p(a)

a,b e S, then

is Noetherian,

S . *

p(a)(b) = p(a

thenthere exists

n e Z+

+ b)

suchthat

P (na) = p(n + l)a) = . . . . o

(2)

If

o

[p]

is nonempty, then

[p]

and[p]

are

o

ideals of

S

and

E

[p]

[p] .

(3)

If p = glb{p^}^ and a e S , then p(a) = n ° n ° glb{p.(a)} ; moreover, [p] = n # [p ] and [p] = l 1 i=l l n

r

i

ni = i [pi ] • Proof.

(1):

compatible with for

The relation + .

holds

Thus,p < p(a) < p(a)(b)

sincep

.

is

Moreover,

s ,t c S , (s,t) € P(a) (b)

iff(s + b)

(s + b + a)p(t + b + a) Hence

p(a)(b) = p(a + b) (2):

a e [p]°

To

show that

and

a + s e [p]

iff

p(a)

(t + b) iff

(s,t)

ep(a + b)

.

.

[p]°

s e S .Then o

so

p ^ p(a)

is an ideal

of

p(a + s) = p(a)(s)

S , take = U(s)= U ,

o

and

[p]

is an ideal.

We observed in

o

Section 1 that

rad([p]

) = [p]

o

taining (3):

[p] For

s,t e S , we have

is an ideal of

S

con-

.

44 sp(a)t

(s + a)

iff

(s + a)P^(t + a) sP^(a)t

that

p(a) = U

that is,

= gib{ p^ (

A n B * <J) , then

we proceed to do.

)

Moreover,

mnx e A n

B , so

.

This equality shows

A

a e n ^[p^]

and

B

» so

[p]

are ideals of

rad (A n B) = rad(A)

ifmx € A

i ---

for each

[p^] , it then suffices, by

The inclusion

is clear.

iff

P^(a) = U

[p] =

induction, to show that if that

a

if and only if

To prove that

i

iff

i .

if and only if

a e [p]

n^[pi]° .

for e a c h

for e a c h

p(a)

Therefore

p (t + a)

S

n rad(B)

; this

rad(A n B) E rad(A)

and

such

n rad(B)

nx e B , then

rad(A) n rad(B) E rad(A

n B) .

This com­

pletes the proof of Theorem 5.5. A congruence

p

primary if, for all p

is prime if

p

is cancellative if

on asemigroup s e S

p * p(s)

S

, p* p(s) implies

p = p(s)

is said to implies that

s e [p]° for each

for

be s e[p]

;

s e S , and

s e S .

It is

clear that a cancellative congruence is prime and that a prime congruence is primary.

A prime congruence

p

is can-

o

cellative

if and only if

p = i

the identity congruence,

only if of

i

is S

[p]

is empty. i

In the case where

is cancellative if and

is a cancellative semigroup, and the definitions

primary and prime closely parallel the way that the

notions of primary and prime ideals are extended from rings to submodules of a module.

If T P t_o

then by the restriction of the relation gruence on

p n (T x T) T .

on

isa subsemigroup of T

, denoted

T ; clearly

p|T

S

,

p| , we mean

is a con­

Theorem 5.6 catalogues some fundamental pr o­

perties of primary, prime, and cancellative congruences.

45

THEOREM 5.6. y,p^,...,pn Let

T

Assume that

are congruences on

be a subsemigroup of

(1)

S

If

y

is

is a semigroup, that S , and that

p = glb{p^}^ .

S .

primary, then

[y]

is a prime ideal of

S . (2) then

If each

p

(3)

If

p

y

[p] = [p^]

.

If each

p^

,

is can­

is also cancellative.

is

primary, prime, or cancellative, then

has the same property. (4)

[y]

is primary with [p^] = ... =[p^]

is primary and

cellative, then

y|^

p^

c

If

y

is

primary, then the restriction of

y

to

is cancellative. Proof.

a,b c S

(1)

Assume that

y

is primary and that

are such that a + b € [y]

n e Z+

so that

x,y e S

.

Then

there exists

(n(a + b) + x) y (n(a + b)

If

(nb + x) y (nb + y)

it follows from the definition of the other

.

case, there exist

(nb + x^,nb + y^)

k y .

Since

x ^»yi

+ y)

for all

x,y e S , then

that

[y]

for all

e ^ such

(na + nb +

b e [y]

.

that y (na + nb + y^ o

and since some

y

m e Z (2)

is primary, , so

Part

(3)

S00 t^iat

such that

p(s) * p .

that p^ *

P^(s)

p

Thus

[y]

is prime in

of Theorem 5.5 shows that

ni i^p *^ = ^Pl^ *

and

mna e [y]

it follows that

a e [y] .

Since

for some

p

P rimaiT>

for S .

[p] = take

s e S

p(s) = glb{p (s)} , it follows i i . Therefore s € [p^] = [p] ,

is primary.

The verification of To prove

(4) , take

(a + x) y (b + x) .

(3)

is routine.

a,b,x e [y]

We must have

Q

In

such that

a y b , for if not, then

46

the definition of

y

primary implies that

trary to assumption. Therefore a y b c of y to [y] is cancellative.

x e [y] , con­

and the restriction

The proof that each congruence on a Noetherian semigroup can be represented as a finite intersection

(greatest lower

bound) of finitely many primary congruences now follows a path that, at least in terminology, mutative ring theory:

is familiar from com­

we define an irreducible congruence,

prove that each congruence on a Noetherian semigroup is a finite intersection of irreducible ones, and that an irre­ ducible congruence is primary. congruence

p

The first step is to define a

on a semigroup

S

to be irreducible if

cannot be expressed as the intersection of two strictly greater congruences on THEOREM 5.7. (1)

S .

Let

S

be a Noetherian semigroup.

Each congruence on

irreducible congruences on (2)

S

is a finite intersection of

S .

Each irreducible congruence on

Proof.

If

(1)

S

is primary.

fails, then among the congruences on

S

that are not expressible as a finite intersection of irre­ ducible congruences, we can choose a maximal one p p

is not irreducible, so < p? .

Maximality of

p = p^ n p^ , where p

implies that each

This establishes

To prove and an element

and

is a

p.

p , a con­

(1) .

(2) , take an irreducible congruence x c S

. Clearly

p < p^

finite intersection of irreducibles, hence so is tradiction.

p

such that p * p ( x )

x € [ p ] . By assumption, there exist

.

u,v e S

P

on

We prove that such that

S

47 Cu,v)

p

but

gruence on

(u

S

p^ < p(x)

generated by

modulo the ideal By Theorem 4.9,

that

nx + S

and

p = pj

.

y

, let of

d p b}

.

Pj z ,

Then

c = nx + c^

and

d

c p(x) y , y p(x) or

p^ = lub{p,y}

By choice of

p (d1 + (n + l)x)

is irreducible,

p

(n + l)x £ nx + S

p

, so

such y Pz

c,d e nx

+ S ,

for some c ,d^

e S .

Therefore

Consequently, so that

^

p( (n +

c^ p(nx)

d^ , yp z

p =

so that

l)x) d^ .

as we

p^ = p^ n p^

holds by definition of p^ .

== lub{p,y}

y £

Since

p

.

(fn + l)x + s) y ((n + l)x + t)

all

s,t e S .

Consequently, C(n + l)x + s) p ((n + l)x + t)

all

s,t e S .

This implies

proof of If sentation

that

be so

c = d , then

z p(x) d .

We have proved that

p^ >

S

c,d € S

p (d^ + nx) = d , and hence

wished to show.

that

y p(x) z

other hand, if

z , and

such

y,z e

.Let

implies

= nx + d^

.

n p^ ; we show

p < p^

n ,this means that

c = (c^ + nx)

c,d e S

Let ,p^ = p^

(c + x) p (d + x) .

(c1 + (n + l)x)

The relation

S , and let

d p z . If

On the

that

be the Rees congruence

Moreover, there exist

and we're finished.

c p(x) d

p^ be the con­

such that

y

y p^ z

that y p c , c y d , and

We have

be

It is clear that

(y + x) p (z + x) .

then

Let

p^ = {(a,b) | there exist

c y d ,

such that

n

+ l)x) = ...

.

p u {(u,v)} ; we note

.Let the integer

p(nx) = p((n

ape,

+ x) p (v + x)

x £ [p]

p Now for for

, completing the

(2) . p

is a congruence on a semigroup p

= p^ n ••• n pn

of congruences on

S

P

S , then a repre­

as a finite intersection

is said to be canonical if the following

conditions are satisfied.

48

and

1)

each pi

is

2)

[P± ] * [P-j]

primary, for

3) pi £ nj ^ipj

The next

i *

j ,

for each

1 *

result follows from Theorem 5.7 and from part

(2)

of

Theorem 5.6. THEOREM 5.8.

Each congruence on a Noetherian semigroup

admits a canonical representation. The proof of Theorem 5.10 uses the fact that the identity congruence on a Noetherian semigroup is a finite intersection of primary congruences.

Before embarking on the proof, we

need to strengthen the conclusion of Theorem 5.3 in the case where the ideal form

[p]

A

.

THEOREM 5.9. ideals, let

p

a finite set

of

Then

in the statement of that result is of the

S = <[p]

°

be

Let

S

be a monoid satisfying

a congruence on

S

, and let

generators for the ideal

a.c.c.

^a i^i=i A = [p]

for ^e

of

S.

c

,a_,...,a ,A > . 1 n

Proof. Let T = . If T = S , we're ----l’ n finished; if not, then an examination of the proof of Theorem 5.3

b e S\T

shows that for

b = b + y .

Thus

b = b + my

, there exists for each

y e A

such that

m e Z+ , and for

m

o

suffiently large, [p]

°

and

S =

<[p]

my e [p] °

,

. Hence

a1 , . . . , am , [ p ]

c

>

S\T is contained

in

in either case.

Before proving Theorem 5.10, we make the trivial obser­ vation that if p is a congruence on a monoid c 0 e [p] if p * U . Since the congruence U

S

, then

is redundant

in any intersection representation with more than one com­

49 ponent, we tacitly assume in Theorem 5.10 that

U

is not one

of the components in the representation of the identity con­ gruence

i (i = U

only if

THEOREM 5.10.

S = {0})

.

A Noetherian monoid is finitely gene­

rated. Proof.

Let

S

be a Noetherian monoid.

We first prove

the theorem under the additional hypothesis that the identity congruence

i

i = p n Pj

on

S

can be represented in the form

n ... n pn , where

is primary.

n = 0 , then

p

is cancellative and each

For this case, we use induction on

i

is cancellative, S

and Corollary 5.4 shows that

S

i = p n p^ n . . . n Pn+1

and let

is finitely generated.

.

n

Let

Case 1.

y = p^ n . . . n Pn+-^

b,y e [y]

such that

[y]

of

show that one of two cases arises.

S = . 1 m ° c S = <[y] ,a^,...am ,[y] > and

Case 2.

As­

and consider the case

be a set of generators for the ideal

S . Theorems 5.3 and 5.9

If

is a cancellative monoid,

sume the result for a given integer where

n .

b = b + y . For

there exist

ksufficiently large,

o

kb

and

ky

are in

[y]

and

kb = kb

+ ky

, so without o

loss of generality we assume that that

[y]°

is cancellative.

b,y e [y]

Thus, take

.

We prove

s,t,x e [y]°

so

o

that

s +

x = t + x .

that

s = (s + 0) y (s + x)

Since

p

s = t

since

By definition of and

[y] ,it follows

t y (t + x) .

is cancellative, we also have

s p t .

Hence

sy t .

Therefore o

cancellative.

i = p n y .

We have proved that

Thus, the equation

is an identity element for

b = b + y

[y]° , and part

[y]

is

implies that (4)

of Theorem

y

50 o

5.1

shows that

[y]

is Noetherian.

We conclude from Corol-

o

lary 5.4 that

[y]

is finitely generated.

Case 1 and Case 2,

S

is generated by

[y]

Thus, c

in both

and a finite n+ 1 [y] = [n^ p.] =

set. Part (3) of Theorem 5.5 shows that n+1 c n +1 c n [p ] so that [y] = u [p.] . Therefore, to show that 1 j 1 J S is finitely generated, it suffices to show that each c [p.] is finitely generated. We examine the identity re3 c n+1 lation on [p^] = A. ; it is n V > where p,|and y = p|A . . By parts j J Theorem 5.6, it follows that and

y

lative.

are cancellative,

y,

k

(3), and (4)

n

of

is primary for

and hence

\i.

n y

Thus, the identity congruence on

section of gruence.

(2),

k * j , y. J is also cancel­ is the inter­

primary congruences and a cancellative con­

Since

Aj = [pj]C

is Noetherian by part (3) of c Theorem 5.1 and since 0 e [p.] , the induction hypothesis c implies that each [pJ is finitely generated. Thus, Theorem 5.10 is proved in the case where

i

admits a primary

decomposition in which one of the component congruences is cancellative. In the general case, let sentative of gruences

i

as

a

i = p^ n ... n p^

be a repre­

finite intersection of primary c on­

(Theorem 5.7).

If

{a.} ™ is a set of generators for o C the ideal [i] , then S = <[i] ,a_,...,a ,[i] > = 1 m ° n c c <[i] , a . . . . , a ,u [p.] > . Each [p.] is a Noetherian 1 m 1 j j monoid in which the identity congruence is the intersection of a cancellative congruence and

n - 1

primary congruences; c by the case of Theorem 5.10 proved above, each [p^] is o

finitely generated. element,

for if

Finally,

c,d e [i]

[i] , then

contains at

most

c + 0 = d + 0 .

one There-

51 fore, the representation shows that

S

S = <[i]

, a^ , . . . ,am ,

[p^. ]c >

is finitely generated.

The converse of Theorem 5.10 is established in Theorem 7.8, which states that a finitely generated monoid is Noetherian. Section 5 Remarks The Primary Decomposition Theorem for congruences on a Noetherian semigroup was proved by Drbohlav [40], who also investigated and proved analogues of classical uniqueness theorems concerning canonical decompositions of ideals of a commutative Noetherian ring.

Drbohlav also transferred his

results from the setting of primary decomposition of con­ gruences to representation of congruence ideals of Noetherian semigroups

(appropriately defined) as a finite intersection

of primary congruence ideals

(also appropriately defined).

The result that a Noetherian

monoid is finitely generated was

proved by Budach [26].

52

§6.

Factorization in Commutative Monoids

Many of the conditions typically considered in the theory of integral domains, such as unique factorization,

are con­

ditions defined in terms of the multiplicative semigroup of nonzero elements of the domain.

In this section we define

some of these concepts in a cancellative monoid and we d e ­ rive some of their basic properties.

Tradition for multipli­

cative notation is strong in the area of factorization, and in this section the semigroup operation is usually written multiplicatively.

Several proofs in the section generalize

familiar proofs in the case of integral domains. Assume that plicatively. write

An

is a cancellative monoid,

element a

of

a|b , if there exists

also say that multiple of a e S

S

a a .

such that

S

divides

c e S

is a factor of

b e S , and we

such that

b

and that

An irreducible element of a = st

in

S

prime if

a|st

ments of

S

are satisfied if

and only if

in which case we

say that

relation on

or

are irreducible.

S

a|t

is a is an element

a = by

aand

b

s

S , and

or a

t is

; as usual, prime ele­

The relations

a|b

and

for some unit

are associates.

y

b|a of S , The

that identifies associate elements is the con­

gruence of Theorem 4.4, where is the group of all units of set of

b

implies that either

(that is, an invertible element) of a|s

b = ac ; we

S

is a unit

implies

written multi-

S , an element

a

H (in the notation of (4.4)) S .

of

X

is a nonempty sub­

is a greatest common divisor

of the elements of

X

each element of

and is divisible by each common divisor

X

of the elements of

X .

(written

S

If

a = gcd{X})

A given subset

X

if

a

divides

need not have a

53

greatest common divisor, but if

gcd{X}

exists, then it is

uniquely determined up to associates.

The definition of a

least common multiple of

lcm{X}

X , written

, is as ex­

pected, and like the greatest common divisor,

lcm{X}

is

determined up to associates if it exists. THEOREM 6.1. monoid, and that (1)

A

lcm{xA}

both exist, then (2)

Assume that

If

and exists

gcd{xA}

If

gcd{A u B}

are nonempty subsets of

if and only if

lcm{A}

exists,

exists;

if

thengcd{A}

exists and

.

a = gcd{A}

and

exists if and

b = gcd{B} exist, then

only if

gcd{a,b}

two greatest common divisors exists, then gcd{a,b} .

S .

lcm{xA} = x • 1cm A

gcd{xA} = x • gcd{A} (3)

B

x,y e S , a cancellative

exists; if these gcd{A u B} =

The analogous statement for least common multiples

is also valid. (4)

If

Moreover,

c = lcm{x,y}

s = gcd{x,y}

Proof.

If

b = lcm{A}

that

c = lcm{xA}

exists, then

gcd{xA} = d , then xA .

exists,

If

(2) d

multiple of

c

is divisible by

if x , say

(1) :

if

is divisible by the common divisor

the proof of and

S .

c^ = lcm{A} .

is similar to that of

d = d^x, then

x

in

straightfor­

xb = lcm{xA} .And conversely,

greatest common divisor of We omit

xy = cs

then it is

It is routine to prove that

The proof of

of

then

.

ward to show

c = c^x .

exists,

y , so

d^

can be shown to

x

be a

A . (3) . xy =

In

(4) , xy

is a common

cs

for some

s e S .

We

54

have

x|c

and

y|c , say

xx^ = c = yy^

cellative, the equalities x = y^s and write

and

t

x = tx2

multiple of tX2y 2 = CZ ‘ s = tz .

x

y-=

and

ty2 .

Clearly

ThSn

Therefore

s= gcd{x,y}

even if

and

tx2y 2

of

y

x and

is acommon tX2^2

* Sa^

t ^S

since

and

.

the converses of

Theorem 6.1 may fail,

x

= SC

is can­

divisor

cdivides

Xy = t2x2y 2 = tCZ

S

imply that

is a common

y ,and hence

It is known that

group of 2 X and

s

be any common divisor of and

Since

xy = xx^s = yy^s

y = x^s ; hence

y . Let

.

(2)

and

(4)

of

S is the multiplicative

nonzero elements of an integral domain. 3 X have greatest common divisor 1 in

semi­

For example, 2 3 Z[X >X ] ,

but

gcd{X3X 2 ,X3X 3 } = gcd{X5 ,X6 } and lcm{X2 ,X3 } fail to 2 3 exist in Z[X ,X ] . Onthe other hand, if each pair of ele­

ments of

S

has a greatest common divisor in

each pair of elements of

S

S , weshow that

has a leastcommon

multiple

in

S . THEOREM 6.2.

Let

x,y e S , a cancellative monoid.

The

following conditions are equivalent. (1)

lcm{x,y} = xy .

(2)

xS n yS = xyS .

(3)

For all

z

in

(4)

For all

z

e S

(5)

gcd{x,y} =1

and

For each

€ S , x|yz implies that

S, x|z

and

y|z

imply that

xy |z .

, gcd{zx,zy} = z . gcd{xz,yz}

exists for each

z e S . (6)

Proof.

z

The equivalence of conditions

the implication

(4) = >

(5)

xlz .

(1) —

are clear, and part

(3) (2)

and of

55

Theorem 6.1 shows that

xyz part

implies both

(4) and

(1) = > (5):

If

lcm{x,y} = xy , then

by part

of

(6.1)

(4)

(1)

of

(2) so that

, and hence

(6) .

lcm{xz,yz}

gcd{xz,yz} = z

by

(6.1) . (6):

z e xS

(6) = > by

(5)

If since

(2):

(6) , x|b

yz = xs , then

If

S

yz e xS n yS = xyS ,

is cancellative.

Therefore

xa = yb g xS n yS , then

so that

b e xS

and

yb e xyS .

xlz .

x|yb , and Consequently,

xS n yS = xyS . COROLLARY 6.3. (1)

The following conditions are equivalent.

Each pair of elements of S

divisor in (2)

S . Each finite subset of

divisor in (3)

S has a greatest common

S . Each pair of elements of S

multiple in (4)

has a greatest common

has a least common

S . Each finite set of elements of

common multiple in Proof.

S

has a least

S .

The implications

(2) z=z> (1)

and

(4) = >

(3)

are obvious, and the reverse implications follow by induction from part implies let

(3) of Theorem 6.1. (1) ; we prove that

d = gcd{x,y} , and write

gcd{x^,y^} = 1 Therefore,

and

Part

(1) implies

(3) .

Pick x,y

e S

,

x * dx^ , y = dy^ . Then

gcd{zx1 ,zy1 >

lcm{x1 ,y1 ) = x 1y 1

(4) of (6.1) shows that(3)

exists for

by Theorem 6.2,and

each z e

S

lcm{x,y}

. =

lcm{dx^ ,dy^} = dx^y1 * This completes the proof of Corollary 6.3. We call a monoid satisfying the equivalent conditions of Corollary 6.3 a

GCD—monoid.

56 THEOREM 6.4. (1)

If

Assume

a,b,c £ S

gcd{a,bc} = gcd{a,dc} (2) If for each (3)

that and

gcd{a^,...,a^} = 1

Proof.

gcd{a,b} = d , then

s

are such that

a^,...,an e S

and

gcd{a^1 (1):

that

t

are positive

=1 .

gcd{a,bc} and

divides

s .

But

(2): where

s = gcd{a,dc}

t

s

gcd{ac,bc} = c • gcd{a,b} = cd , and hence Therefore,

gcd{a,b^} = 1

are such that

k^,...,kn

,...,a^n }

Let s =

It is clear that

GCD—monoid.

gcd{a,b1b 2 ...bR } = 1 .

Assume that

integers.Then

is a

.

a,bj_,...,bn e

i , then

S

= gcd{a,dc} divides

s

divides t

By induction, it suffices to consider the case

n = 2 , and this case follows immediately from

(3):

of

(1):

1 .

It is clear that we need only consider the case

k_ = ... = k _ = 1 . 1 n-1

Moreover, since

gcd{gcd{a in

.

.

gcd{a,b1b 2 > = gcd{a,1 • b 2 > = gcd{a,b2 > =

where

.

,...,a },a } , it suffices toprove 1 n-1 n (2) for n = 2 . This case, however, is a

gcd{a_,...,a }= i n the

statement

consequence

(2) . Unique factorization into primes can be defined in a

cancellative monoid either in terms of elements or in terms of principal prime ideals.

We choose the latter approach

here because statements of results are a bit cleaner in the language of ideals. easy, being

based

and

sS

only if

Transfer between elements and ideals is on three facts:

is a

(i)

prime ideal of

irreducible if and only if either

s

se S

is prime if

S ; (ii) is a unit of

s e S S

or

is sS

57 is maximal in the set of proper principal ideals of (iii)

sS = tS

if and only if

s

and

t

S ; and

are associates.

We say that the cancellative monoid

S

is factorial

(or

a unique factorization monoid, abbreviated UFM) if each prin­ cipal ideal of

S

can be

principal prime

ideals of

written as a finite product of S.

The next result shows that

such factorization is unique to within order and factors of S . THEOREM 6.5.

Let

S

be a cancellative monoid that con­

tains a proper principal prime ideal.

Let

P

be the

multiplicative semigroup consisting of all principal ideals of

S

that are expressible as a finite product of proper

principal prime (1)

If

ideals of

xS € P ,

S.

then xS

is uniquely expressible,

to

within order, as a finite product of proper principal prime ideals of

S .

(2) If

e

xS =

prime ideal of

p^

e

1

S and

k ^ » w here each

p.S * p.S i

{pi

(3)

If

of S

S

i* j

is a ,

for each i}

containing

xS .

is factorial, then

p^p2 ...pnS > S .

To prove

(1) , let

where each

S

satisfies

p^

x

is prime; if xS =

then

x^ e xS

or

x2

exS

the as­

(a.c.c.p.).

xS =p^Sp2S...p

S =

is a nonunit prime element of

We obtain uniqueness by induction on

then

then

is the set of prin­

cending chain condition for principal ideals Proof.

proper

j

'--P^ S | 0 ^ f^ ^ e^

cipal ideals

for

p^S

x^S • x 2S = so that

xS

n .

If

n = 1 ,

in this case, = x^S or

xS

= x 2S .

58

If

xS = x^S , then

x 2S = S

since

Thus, factorization is unique if factorization

for n = k

S

is cancellative.

n = 1 .

We assume unique

and consider the case where

xS = p ...p p S . Assume that xS = q S • ... • q S 1 k k+1 1 m another factorization of xS into proper prime ideals. p^S 2 q_. S

for some

j —

s

canceling

in the equality

1

Then

p^S o q^S .Writing q^ = p^s ,

say

it follows that p

is

is a unit

of

S

and

pS = q S . 1 1 p ...p S = q ...q S 1 k+1 1 m

P i - - - P w i s = tlo---cI s . whence k = m - 1 and / J K + l z m for 2 < i < m by proper labeling. Thus (1)

Then yields

p.S = q.S follows by

the principle of mathematical induction. The proof of if

(2)

follows by the same method as in

e = e^+...+e^ , then we use induction on

that each principal ideal of required form. in

For

ideal* of

S

Therefore have p,y..S 11

holds

n + 1,

containing

y

e p^S

containing xS

is of the

fore = n , then in the case

let yS

xS .

y^S 2 p®l'*...p®ks

be

a proper principal

We write

orz e p.^S .

If

x = yz

y =

p^y^

e p^S

0 < f. < e. i i

i , follows from the induction hypothesis.

.

, then we

and the representation of

h fk p., ...p, S , with 1 k

in the form

to prove

e = 1 , the proof is essentially given

(1) .If the result

where e^+...+ek =

S

e

(1) :

yS = for each

Similarly,

if

e-, -1 ev Bp ...p S and the desired conclusion 1 l k again follows from the induction hypothesis. z e p

S,

then

(3):

If

yS

S

is factorial, then

finitely many principal ideals of cipal ideal

of

S

.

Thus,

S

a.c.c.p.

(2)

shows that only

contain

a fixed prin­

is satisfied in

S .

59

COROLLARY 6.6.

Assume that

the factorial semigroup

iff

X

is a nonempty subset of

S .

(1)

gcd{X}

exists

in S

(2)

lcm{X}

exists

in Siff {xS | x e

the elements of Proof.

Take

p S,...,pn S of

X

eS

and nonnegative

Theorem 4.5 implies that c c p,^...p n

i .

*i

Thus,

S

of

S

X

X .

ideals of

If

S

of

integers

S , where

S .

contain

that contain dS .

elements of

X .

X , then

ideal

If

e = lcm{c,d}

.

c

of

for each

’i1 ’i

exists

for each

the case of an arbitrary

y

d

X , there exists

e

a

is a common divisor of the

is any common divisor of the elements for each

y

e

yS .

X .

Consequently, the

But since

S

is principal, generated by

By choice of

ds n cS c cS , and hence

set

(2)

= inf(a.,b }

gcd{X}

for each

contains each

dS n cS

multiple

Part

Hence, among the principal ideals

The element

GCEMnonoid,

(2):

.

such that

have greatest common

c

Consider

xS . yS

yS s cS

dS n cS

d = gcd{X}

s^

and

primes

x e X , then only finitely many principal

minimal one

of

s^

is aGCDr-monoid and

finite subset subset

in

*n

is finite

andchoose principal

.. p N ;

divisor

S)

have a common multiple.

s^, s

S

.

c

is a e , where

dS , it follows that divides

d .

dS = eS =

This proves that

. Assume that the elements of m .

{xS | x e

where

x. e X

Since

S

i

is a

Then X}

mS c xS

for each

is finite, say

for each

i .

GCD-monoid,

Then

X

have a common

x e S , and hence the

{xS | x e S}

=

»

n1? ..x.S = n{xS I x e X} . i=l l 1 nl=ix is is principal, generated

60

by some

t € S .

The equality

tS = n(xS | x e X)

alent, however, to the assertion that Theorem 6.5 and

Corollary 6.6 show that if

factorial semigroup, a

GCD—monoid.

t = lcm{X}

then

S

satisfies

is equiv­ . S

a.c.c.p.

is

a

and

S

is

It is well known from the theory of integral

domains, however, that neither tion implies factoriality.

a.c.c.p.

nor the

GCD—condi­

On the other hand, we show that

the combination of these two conditions does imply factor­ iality.

. On the other hand, we show that the combination

these two conditions THEOREM 6.7. (1)

If

S

does imply factoriality.

Assume that satisfies

cipal ideal of

S

of

S

is a cancellative monoid.

a.c.c.p.

, then each proper prin­

can be expressed as a finite product of

maximal proper principal ideals of

S ; equivalently,

each

nonunit of

S

is a finite product of irreducible elements of

S .

S

is factorial if and only if

Hence

a.c.c.p.

and irreducible elements of

(2) of

S

If

S

is a

S

S

satisfies

are prime.

GCD— monoid, then irreducible elements

are prime. (3)

S

a.c.c.p.

is factorial if and only if

and

Proof.

S If

is a S

S

satisfies

GCIMnonoid.

satisfies

proper principal ideals of

S

a.c.c.p.

, then the set of

admits maximal elements;

such

an element is what is meant by a maximal proper principal ideal. of

S

If

(1)

not expressible as a finite product of maximal ones,

there exists one xS

fails, then among the proper principal ideals

xS

that is maximal within the set.

is not a maxiaml proper principal ideal of

Then

S , so there

61 exists

y

where yS

e S

z

such that xS

< yS

< S

.We write

is necessarily a nonunit of

and

zS

S.

x =

yz ,

Then each of

is a finite product of maximal proper principal

ideals, hence so is

xS .

This contradiction establishes

CD • (2):

Assume that

is irreducible. wise, pick so part

s

is a

GCD— monoid and that

is a unit, then

x,y e S\sS .

(2)

Therefore

If

S

Then

s

4 sS

and sS

is prime.

Other­

gcd{x,s} = gcd{y,s} = 1 ,

of Theorem 6.4shows that

xy

s e S

gcd{xy,s} = 1

.

is prime as asserted.

We have already observed that a factorial semigroup sat­ isfies from

a.c.c.p. (1)

Let

and S

of units of

and is a

be a factorial semigroup and let S .

If

(p„)„ A cl

P

a

n °° = {p } a n=0

S

is the internal weak direct sum of subsemigroups of

for

each • G ©

to a monoid of the form each

Z

a

a w

.

Thus,

> where

S G

is isomorphic to the additive monoid ^

negative integers.

be the group

is a complete set of nonasso-

a £A

and of the family

where

G

S , then the very definition of

factoriality implies that G

the converse follows

(2) .

ciate prime elements of

of

GCD—monoid;

s *

is isomorphic r

is a group and 0

of non-

Clearly each monoid of the latter form is

factorial, so we have established the following theorem. THEOREM 6.8.

Let

G

be a group, let

^e a

family of monoids, each isomorphic to the additive monoid -of • w nonnegative integers. The monoid G © Ea Za is factorial. Conversely, each factorial monoid a monoid

G © E Z a a

S

is isomorphic to such

62 We note

that

G

and

A , determine the monoid for

G

|A|

,

the cardinality of the set

G © ^aeAZa

t0 w ^thin isomorphism,

is the group of units of the monoid and

|A|

is the

cardinality of a complete set of its nonassociate prime ele­ ments.

Is there anything special about the case where

is finite?

Based on w ha t’s true for integral domains

|A| (a

factorial domain with only finitely many nonassociate prime elements is a principal ideal domain), we might conjecture that each ideal of the monoid is principal if but this statement is false.

For example,

|a |

is finite,

S = ZQ © Z^

is

factorial with exactly two nonassociate primes, but the ideal + * + Z © Z of S is not principal.

Chapter II

SEMIGROUP RINGS AND THEIR DISTINGUISHED ELEMENTS

This chapter contains five sections.

In the first of

these, Section 7, semigroup rings are defined and some of their basic properties are established.

In particular,

Theorem 7.7 is the result that a unitary monoid ring R[X;S]

is Noetherian if and only if

is finitely generated.

is Noetherian and

S

Subsequent sections treat zero divi­

sors, nilpotent elements, rings.

R

idempotents,

and units of semigroup

While each of these classes of elements is of interest

in its own right, it is also the case that knowledge of these classes of distinguished elements is frequently an indis­ pensable tool in work on global structure properties of semigroup rings in Chapters III and IV.

63

64 §7.

Semigroup Rings

Apart from the definition of a semigroup in Section 1, all semigroups considered in the main text up to this point have been assumed to be commutative.

We intend to continue

to observe that convention, as well as to assume that all rings considered are commutative. nition of the semigroup ring of

But in giving the defi­ S

over

R , it seems

pointless to place such restrictions on the semigroup the ring

R .

assume that semigroup. R

S

or

Thus, in this first paragraph of Section 7, we

R

is an associative ring and that

Let

T

be the set of functions

f

(S,*) from

is a S

into

that are finitely nonzero, with addition and multiplica­

tion defined in

T

as follows.

(f

+ g)(s) = £(s) +

gfs)

(fg) (s) = Zf(t)g(u) t*U=5 where the symbol over

E indicates that the sum is taken t*u=s (t,u) of elements of S such that

all pairs

t*u = s ,

and it is understood that

not expressible in the form fication

that

T

the special case familiar: R[X]

for

,

t*u

+ and

S = ZQ , T

is

Ve ri ­

•follows as in

is simply the polynomial ring R .

R ; to denote

We call

ring of

S

over

or

.

If the semigroup operation on

T

T

the semigroup

, we use either S

multiplicatively, we use the notation

R [S ]

elements of

£

T

s

, with which the reader is no doubt

in one variable over

R[S]

if

t,u e S .

for any

is a ring under S =

(fg) (s) - 0

are written either as

R[X;S]

is written for

f(s)s

T

, and

or as

65 Si= l ^ si^si * Thus> S is a free basis for ule, and the multiplication in utivity and by setting operation in

S

R[S]

R[S]

as an

R—m o d ­

is determined by distrib-

r^s^ • r2 s2 = r^r2s^s2 . If the semigroup

is written as

ten either in the form

+ , the elements of

E cf(s)Xs or in the form seS

T

are writ­

E? .f (s . ) ^ , i=l 1' *

with addition and multiplication defined as for polynomials. Chapter 2 deals primarily with properties of elements of the semigroup ring

T , and

here to denote

T

.

R[X;S]

is the symbol customarily used

Subsequent chapters deal, however, with

global properties of semigroup rings, there is much less e m ­ phasis on the form of elements of we use

R[S]

notation

X

to denote

T . Introduction of the symbol X and the

has the effect of transforming

multiplicative semigroup phism r

T , and in Chapters III—V

s -- ► X s

{Xs | s e S}

(S,+)

into the

by means of the isomor­

. Nevertheless, the notation

E^ ,f.Xsi i= l i

is more commonly used in commutative ring theory than Y

E^_^f^s^ , no doubt because of customary additive notation in commutative semigroups and because of the suggested analogy between semigroup rings and polynomial rings. degree to which

R[X]

serves as a reasonable model for

depends partly upon properties that For example, to

R[X]

R[X^,...,X ]

R[{Y }

.]

A A €A

R[X;S]

S has in common with

Zq .

serves as a good model for generalization for some ring— theoretic questions, but as a

poor model for certain other questions. ring

Of course, the

over

R

Yet each polynomial

in commuting indeterminates

Y,

A

can easily be seen to be isomorphic to a semigroup ring over R , as follows. A , and let is, the

Ath

Let

F = e Y AZ. , where A eA A

Z. * Zn for each A O

be the standard free basis for coordinate of

coordinates are

0 .

e

A

is

1

F —

that

and all the other

66

Each element of

F

is uniquely expressible in the form

Ek e for some k > 0 , and identification of A A A i a kx r X e R[X;F] with r nY, e R[(Y,}] , with sums a a a a in

R[X;F]

a =

a E

r X aeF a

identified with the corresponding sums in

R C (Y x }] , is easily seen to define an isomorphism of R[X;F]

onto

R[iY.}]

.

A

In the case of a finite set {Y.}?

1 1— 1

of indeterminates, isomorphism of R[X;Z^]

R[Y , . . . ,Y ] and 1 n follows by induction from the next result.

THEOREM 7.1. that

S

and

R[X;S & T]

T

Assume that

is a commutative ring and

are additive abelian semigroups.

is isomorphic to

semigroup ring of Proof.

R

T

Then

U = (R[X;S]) ([X,T]) , the

over the semigroup ring of

S

over

R .

Throughout this proof we regard elements of a

semigroup ring as finitely nonzero functions from the semi­ group into the ring, as defined above. <J> : U -- * R[X;S © T]

as follows.

(s,t) e S @ T

[<j>(f) ] (s,t) = [f(t)](s)

, then

If

Thus, define

is well-defined since each element of function from

T

jective, for if defined by

into he

.

R[X; S © T]

[f(t)](s) = h(s,t)

f * g , then

f(t) * g(t)

[f ft)] (s) * [g(t)](s) <j>(f) * (g) •

U

.

The map

Moreover,

is sur-

, then the element

for some

Thus, if



is a well-defined

is such that

for some

and

t e S .

f e U

4>(f) = h .

If

Therefore

s e S , and hence

To complete the proof, we show that

ring homomorphism. then

R[X;S]

f e U

f,g e U

and

<J>

is a

(s,t) e S © T ,

67

[
= [ (£ + g)(t)](s)

= [f(t)

[£(t ) ] (s) + [g(t)] Cs) = [♦Cf)](s,t) +

[4>C£)

so that

+ ifr(t) ] (s,t)

<(i(£ + g) = (f) +
[+(£g)](s,t)

= [fg(t)](s)

=

- U = t (Ec d = s ' f(a)1(c!

♦ g(t)](s)

[♦(g)Hs.t)

Moreover,

tEa+b = t f(a)g(b)](s)

• tsCb) ] (d)) =

4>(fg)

=

H c , a ) }{[*Cg)] (d,b) } =

[♦(f)(g) ] Cs ,t)

and this shows that

=

,

E ( c , a ) +( d,b)=(s,t){ t£ ^ H c ) } a g C b ) ] ( d ) }

E (c,a)+(d,b)=(s,t)*

=

=

,

•

Therefore

<J>

is an

isomorphism and the proof is complete. Henceforth we assume that all semigroups and rings con­ sidered are commutative, and we are not hesitant to restrict consideration to monoids and/or commutative unitary rings, for this is the arena with the most interesting action. S

is a monoid, we sometimes refer to

ring of

S

in a monoid

over

R .

S^ = S

Since each semigroup

u (0)

as the monoid S

is imbeddable

, restriction to monoids frequently

amounts to no loss of generality. R

R[X;S]

If

admits unital extension rings

Similarly, since each ring R[el

whose structure is

generally well known in relation to that of

R , the hypo­

thesis that a given coefficient ring is unitary may amount to a matter of convenience, rather than a real restriction. On the other hand, the presence of an identity element in

R

68 plays essentially no role in most of the problems considered in Chapter 2, and in such problems, the unitary hypothesis on

R

A[X;T]

is not automatically imposed.

Semigroup rings

, where

T

A

naturally arise, ring If

R S

or if

for example, if T

R[X;S°]

f(0) = 0 .

if

R , then

R* = R[e] R [ X ;S ]

consisting of all functions s e S .

with

e R[X;S] , and consider R

rX° Each

in the form

the support of

f

R

is the ideal of f

such that

as a

of R[X;S]

I^_^f^XS * , where

f .

The subset

f , and

f(s) € R r e R

subring of

R[X;S]

has a unique repre­ f^ * 0

and

f

c(f)

S^

Supp(f)

.

The

is called the content of .

It is clear that

for a finitely generated subring generated subsemigroup

is called

is denoted by

generated by

and is denoted by

^si^iS

i-s the supporting semigroup

The support of

ideal of

is a unital ex­

is a monoid, we identify

f

such

i x j ; this representation is called the

for

canonical form of

f .

S

nonzero element

sentation

of

If

S .

can be considered as

consisting of all f e R[X;S°]

for each

s^ * Sj

is an ideal of a unitary

R[X;S]

Similarly,

tension of the ring R*[X;S]

A

is not a monoid, may

is a subsemigroup or ideal of a monoid

is not a monoid, then

the ideal of that

is not unitary or

of

R^ S .

of

R

f

e

R^fXjS^]

and a finitely

Since finitely gene­

rated rings and finitely generated monoids are Noetherian, certain questions concerning units, potent elements of

R[X;S]

idempotents, and nil-

considered in the rest of this

chapter can be reduced to the case where

R

and

S

Noetherian The next result contains some basic information

are

.

69

concerning homomorphisms of semigroup rings. THEOREM 7.2. ring

Rq , let

A

homomorphism of mappings

Let

be a homomorphism of

y

be the kernel of

on

t

: R[X;S] -- - R Q [X;S]

y*

y , and let

S into the semigroup

y* » 4>* » and

Sq

R[X;S] ; V* (

R

Z

.

as ^

into the <J>

be a

Define the follows.

r

= z"y(r i )XSl

* : R[X;S] -- ► R [ X ; S 0 ] ; * * ( 2 r . X S i) = S r . X * (Si)

: R[X; S] -- >- R 0 [X;S0] ; i C E r ^ i )

t

,

,

= Ey (ri)X
.

Then the following statements hold. (1)

y*

surjective if (2) I

of

<J>*

R[X;S]

is a homomorphism with kernel y

t

are

generated by
{rXa — rX*5 | <J>(a) = <J>(b)} ;

is a homomorphism with kernel

It is

and

y

routine toverify that

, <J>* , and

y*

three maps.

of

We verify the equality

That

<j)*CrXa - rXb ) = rX<*’^a ') - rXl)>('b ') = 0 follows that

I E ker 4>* .

take a nonzero element to see that and

A[X;S] + I ; t

are surjective.

t

jectivity of these is clear.

m > 1

<J>*

is surjective.

homomorphisms, as well as the statements concerning

y*

is

is surjective.

is surjective if Proof.

; y*

is a homomorphism and its kernel is the ideal

is surjective if (3)

A[X;S]

.

fj = -f2 , so f

if

is the

sur-

kernel

I = ker <J>*.

Since

(b) , it

To prove the reverse inclusion,

f If

A[X;S]

t

e ker <J>* .

m = 2 , then clearly

= fjX*51 - fxX S2 € I .

there exist distinct elements

si»sj

°f

I t ’s easy (s^) =

Cs 2 D

m > 2

,

For

Supp(f)

such that

70

* By induction, 1= ker * . and

Thus,

(r^XS i — r^XSj)

with fewer than g

is in

m

is an ele­

elements in its support.

I , and hence so is

The assertion concerning

f .

ker t

Therefore

follows from

(1)

(2) . COROLLARY 7.3.

R

g = f —

and that

I = ({rXa —

Assume that

~ is

A

is an ideal of the ring

a congruence on thesemigroup | re

rX^

R

anda ~ b})

,

, and

R [ X ;S]/ (A[X;S]+I) ~ (R/A)[X;S/~] The ideal

I

Theorem 7.2 shows that

If the ring

.

of Corollary 7.3 is called the kernel

ideal of the congruence

elements of

. Let

.Then

R[X;S]/A[X;S] ~ (R/A)[X;S] R [ X ;S ]/1 ~ R[X;S/~]

S

~ . I

a form

the

The proof of part

(2)

of

consists of all finite sums of b

rX

R isunitary,

— rX

then

course, a set of generators for

, where

r e

Randa

(xa — X^ |a ~ b> is,

I .

~b.

of

Ideals of the semi­

group ring that are kernel ideals have some interesting properties, as we proceed to show. THEOREM 7.4. the ring

R

and

on the semigroup (i) Moreover, fied.

Assume that I

A,

is the kernel ideal of the congruence

S .

nAeA{A A [X;S]+I} = ( n ^ ) if

R

are ideals of

is unitary,

then

[X;S] + I . (2) and

(3)

are satis­

~

71 A [X;S] n (B[X;S]+I) = (A n

(3)

Proof.

(1):

homomorphism

of

to theassertion thisequality (2):

Since

I

R[X;S] that

B)[X;S]

+ AI.

is the kernel of

onto

the

[R X;S/~] , (1)

canonical

is equivalent

n(A^[X;S/~]) = (nA^)[X;S/~]

,and

is clearly satisfied.

Since

R

is unitary,

A[X;S] = A*R[X;S]

Thus

(A[X;S])•I = AI .

The inclusion

patent.

of the reverse inclusion is similar to the

The proof

proof of

(2)

in Theorem 7.2.

f = ^i=i^ix S ^ 6 1 n A[X;S] and

m = 2

implies

.

AI

.

Thus, Then

distinct

s ± > sj

moreover,

e A [ X ;S]

implies that

(3):

e AI

In view

f^ e

f e AI .

A

For

€ Supp(f) such that

, we have of

m > 1 , and m > 2 , si ~ sj >

f^ e A .The induction

s* s• f — f^(X 1 — X J) e AI , and since

hypothesis implies that X 5^

implies

f = f^CX5! — Xs2) > where

there exists

f^(XS;*-

take a nonzero element

f eI

s^ ~ s^ , from which it follows that

f

£ (A[X;S]) nI is

f e AI

(1) and

.

(2) , assertion

(3)amounts

to saying that intersection distributes over sum in This is known to hold if

B c A .

(3) .

But we can reduce to that

case since A [ X ;S] n (B[X;S]+I)

=

A [X;S] n (A[X;S]+I)

n(B[X;S]+I)

A [ X ;S] n { (A n B)[X;S]+I>

=

.

This completes the proof of Theorem 7.4. Theorem 7.5 establishes a close relationship between the lattice of congruences on

S

and the set of kernel ideals

72 of

R[X;S]

and

.

In the statement of Theorem 7.5, we use

, respectively, to denote the least upper bound and

a

greatest lower bound of two congruences on THEOREM 7.5. group.

Let

With each

kernel ideal

R

congruence on

I

of

ideals of

R[X;S]

I .

S

into the lattice of

has the following properties.

is injective.

(2)

0

preserves order — that is,

p1 <

p2

if and

I (p1) c I (p2) . If

and

p2

arecongruences

ICPi a P2) £ ICP^D n I (P2^ *» R

onS , then

inclusion may be proper.

Moreover,

if

and

(5)

are satisfied.

(4)

I(Pl v p2) = I(px) + ICp2)

is unitary and S

is a monoid,

then

for congruences

(4)

P 1 >P2

S . (5)

The congruence

only if the ideal { (s jt.)}11 i i i=l generates I Proof. S ,

be a semi­

: ~ ------ ► I(~)

The mapping 0

0

(3)

S

S ,associate the

(1)

only if

S .

be aring and let

from the lattice of congruences on

on

v

I (~)

~

is finitely generated if and

is finitely generated.

generates ~ if and only

if

If

p^ and

p2

are distinct congruences

sp^t

while

but

not in ICP2D •This proves

(s,t)

p^ < p^

{ p2 .

Then

Xs (1) .

s,t e Xt

the fact that

a {rX

I Cp -j^D •

inclusion I(p

A p 2) c

That

b - rX| ap-^b) I(p1)

S

such

is in ICP^D

is clear, and the other part

follows from The

{XSi — X**}1? i=l

•

then wemay assume that there exist

implies

In fact,

on that ,

ICP^D £ I ^P2^ of

(2)

generates

n I( p 2)

in

(3)

73 follows from

(2) .

Part

(3)

the inclusion may be proper. I (p1) + I ^p2^ - ^ pi V p2 ^

of Theorem 7.6

implies

that

The inclusion

in

^

also follows from

(2) .

For the reverse inclusion, it suffices to show that a b X - X e Ifp^) + IC P 2 ^ f°r each (a,b) e p^ v p2 . From the description of

p^ v p2

that there exist elements

given in Section 4, it follows s^,s2 ,...,sn e S

such that

a = Sj.b = sn ,s1p1s2 ,s2p2s3 ,s3p1s4 ,...,sn l p2sn .

Then

xa -

XSi+1

in

Xb = Z)"Ji(XSi -

I(p1) + I(p2) .

and the equality

XSi+1) , where each Xa -

Therefore

XSi -

Xb £ Ifp1) + K

I (p^ v p2) = I(p^) + 1 CP2)

p 2)

then follows.

It is clear from the second statement that

For the converse, note that if then the generating set

{X

a

I(~) I

-

X

i

b

s

finitely generated,

| a ~ b}

for

I(~)

proceed to prove the second statement.

generated by clear. if

~ .

Let

{XSi -

•

We define a congruence

Xs -

X^ e J

.

Clearly

each

i , ~ < y , and hence

that

I(~) = I(y)

y

y

J = Ify)

I

c

{XS i -

be the congruence on

We have just proved that

I (y)

S

is

(5) .

We

S .

R [ X ;S ]

J c I (~)

by setting Since

{XS i —

generated by is generated by

I

for

It follows

X*^}^ .

generates

is

s y t

s^ y t^

I(y) s I(~) .

X^i}^

~

be the ideal of

on

con­

Thus, assume that

The inclusion

is generated by

versely, assume that let

J

I(~)

, whence

finitely generated by the second statement in

generates

~

is finitely generated.

tains a finite set of generators for

)}^

(5)

We observe that the first statement

finitely generated implies that

(Cs^t

,

established.

Assume for the moment that the second statement in has been established.

is

Con­

and n {(s^,t^)}^ .

74

{XS i y = ~

X t i)1 , and hence and that

~

I fy) = I (~)

is generated by

» implying that

{(s^,t^)}^

.

Theorem 7.5 has a number of important consequences. fore looking at some of these, we restate case where on

G

S = G

is a group.

correspond to the subgroups, andG/~ , for

is unitary,

generated by 1 —

the corresponding ideal {1 -

X^ e I (~)

X*1 | h e H}

{1 -

Xa | a e A}

~

H , is I(~)

the G/H .

of

R[X;G]

ifg e H .For a subset

A generates

generates

If is

in thiscase, and

if and only

G , it followseasily that

in the

In that case, the congruences

congruence corresponding to the subgroup R

(7.5)

Be­

I (~) .

H

A

of

if and only

if

These observations, to ­

gether with Theorem 7.5, yield the following result. THEOREM 7.6. Define a mapping

Let <J>

R

be a ring and let

G

be a group.

from the lattice of subgroups of

into the lattice of ideals of

R[X;G]

a subgroup of

is the kernel of the canonical

map from

G , then

R[X;G]

onto

(H)

R[X;G/H]

as follows.

G

. The mapping

If



H

is

has the

following properties. (1) (2) only if (3)

is injective. For subgroups

of

G , H^ c H 2

if and

(H^) c 4>(H2) . <|>(H

n H 2) c

if and only if Moreover,

H ^ , H^

if

^(H ) n

H^ s H 2 R

or

<j>(H2) , with equality holding

H2 “ H1

is unitary, then

’

(4)

and

(5)

are satis­

fied. (4)

d>(Hi

+ H 2) =

(5)

The subgroup

<*>0^) +

<J>(H2) •

H

G

of

isfinitely generated

if

75 and only if

<J>(H) is finitely generated.

set

Ggenerates

generates


Proof.

implies ^

r £

R.

in

c H2

and only if

£

(3)

that

{1

sub­

-

X^}^

<J>(H^ n

or

and choose

Then

g e H 2^H 1 *

= rX° -

H 2) =

requires proof. Let ^

rXh+ rXh+g -

hence in (J>(H^ n H 2) .

(H2),

hence

fact, a

.

Assume that

are not

if

Only the assertion in

n 4>(H2)

and let

H

In

n H 2 ,it follows that

Since

rXg

6 Hl is

h + g

h e H^

in

and

g

n H 2 , and

H, c H . 1 “ 2 We remark that the lattice of subgroups of the nontriv­

ial abelian group

G

is linearly ordered if

and only if

G

is cyclic of prime-power order or a quasicyclic group (Theorem 19.3), and hence examples where the inclusion in (3)

is proper abound.

S x S

on

S , then

If

S/~

~

is the universal congruence

is the semigroup with only one

element and the semigroup ring R . of

R[X;S/~]

is isomorphic to

We call the corresponding homomorphism R[X;S]

kernel

onto

I

R

the augmentation map on

R[X;S]

is called the augmentation ideal of

is the ideal generated by THEOREM 7.7. monoid.

I^r^XS i -- ^ l r i

Let

R

{rXa -

rX*5 | r e R

; its

R[X;S] and

, and

a,b e S} .

be a unitary ring and let

S

be a

The following conditions are equivalent.

(1)

The monoid ring

(2)

R

Proof.

R[X;S]

is Noetherian and Assume that

is a homomorphic image of

S

R[X;S] R[X;S]

is Noetherian. is finitely generated. is Noetherian.

Since

R

, it is also Noetherian.

76 Moreover,

R[X;S]

satisfies a.c.c.

Theorem 7.5 implies that —

that is,

S

S

on congruence ideals, so

satisfies

is Noetherian.

Thus,

a.c.c. S

on congruences

is finitely gene­

rated by Theorem 5.10. Conversely,

if

R

is Noetherian and

is finitely generated, then

S =

<s1 ,...,sn >)

R[X;S] = R[Xs 1 ,...,XS n ]

finitely generated ring extension of

is a

R , and hence is a

Noetherian ring. We remark that the case of Theorem 7.7 where

S

cancellative is much easier than Theorem 7.7 itself.

is This is

so because the proof that a cancellative Noetherian monoid is finitely generated (Corollary 5.4) is less profound than the proof of the general case

(Theorem 5.10).

The next result is

the converse of Theorem 5.10. THEOREM 7.8. Proof.

Let

monoid ring

A finitely generated monoid is Noetherian. S

be a finitely generated monoid.

Z[X;S]

The

is Noetherian by Theorem 7.7, and this

implies, by Theorem 7.5, that

S

satisfies

a.c.c.

on

congruences. A monoid

S

is said to be finitely presented

finitely definable) if rated free monoid p

on

F .

F

S ~ F/p

(or

for some finitely gene­

and some finitely generated congruence

The next result is a theorem of Redei

[122,

Theorem 72]. THEOREM 7.9. presented.

A finitely generated monoid is finitely

77 Proof. then

S

If

S = <s^,...,s

is finitely generated,

is the homomorphic image of the free monoid

Z q ; hence

S - F/p

shows that

p

for an appropriate

F =

p , and Theorem 7.8

is finitely generated.

Note that since a free monoid is cancellative, the proof of Theorem 7.9 can be obtained without appeal to any of the material in Section 5 beyond Corollary 5.4. For semigroups that need not be monoids, Theorem 7.10 gives partial results concerning conditions under which a semigroup ring is Noetherian. THEOREM 7.10.

Let

R

be a ring and let

S

be a semi­

group. (1)

If S

and only if (2)

R

is finite, then

R[X;S]

is Noetherian

if

R[X;S] is Noetherian,

then

is Noetherian.

If S

is infinite and

R

is a Noetherian unitary ring; the convere fails, even for

S

finitely generated. Proof.

Since

R

is a homomomorphic image of

the Noetherian property in Moreover, R[X;S] so

if R

In R 2

is Noetherian and

is inherited by S

R .

R—module,

is also Noetherian as a ring.

(2) , we first prove that is unitary.

(R/R )[X;S]

,

is finite, then

is a finitely generated (hence Noetherian)

R[X;S]

that

R[X;S]

R[X;S]

R [ X ;S ]

Noetherian implies

Consider the Noetherian ring

.

Since multiplication in this ring is trivial, 2 the additive group of (R/R )[X;S] is Noetherian, hence finitely generated. additive group of

It is also true, however, that the (R/R )[X;S]

is the weak direct sum of

78 2 |S|

copies of the additive group of

that

R/R2 = {0} , so

then follows that converse, R[X;Z R

]

R

R = R2 .

R/R

Since

.

R

This implies

is Noetherian,

has an identity element.

the proof of part

(2)

it

For the

of Theorem 20.7 shows that

is Noetherian if and only if the additive group of

is finitely generated. The final result of this section is in the same vein as

Theorem 7.9 in that it describes all monoid rings over a fixed unitary ring free monoid rings

R

as certain homomorphic images of the

(that is, polynomial rings) over

R .

For

the statement of Theorem 7.11, we introduce the ad hoc terminology pure difference binomial for a polynomial g e R[{Xa >]

of the form

g = X°1...X*£ -

Let

for

•

some nonnegative integers THEOREM 7.11.

xjl...xjn

R

be a unitary ring.

isomorphism, the class o*f monoid rings over

To within

R

can be char­

acterized as the class of all residue class rings J/A

A A eA

, where

A

is an ideal of

by pure difference binomials. monoid, then

R [ X ;S ]

ring

If

S

R[{X 1] A

generated

is a finitely generated

is isomorphic to such a residue class

* where

A

is finite and

A

is generated

by a finite set of pure difference binomials. Proof.

Theorem

sults already proved let

S

S .

If

each F .

7.11 is primarily a translation of re ­ into slightly different language.

be a monoid and let { s ^ X e A F

is the free monoid

A , then

S ~ F/p

E™e^Z^

Thus,

a Seneratin S set f°r , where

Z^ - ZQ

for an appropriate congruence

But Theorem 7.2 shows that the kernel

I

of the

for p

on

79 canonical homomorphism of rated by

(Xg -

R [X ;F ] onto

R[X;F/~]

is gene­

X^|g ~ h} ; moreover, under the canonical

isomorphism of R[X;F] onto R[{X x >x €a ] » the elements a h X6 — X are precisely the elements of R[X;F] that map to pure difference binomials in each monoid ring over A

R

R[(X^}]

.

This proves that

is of the form

R[{X^}]/A , where

is generated by pure difference binomials.

finitely generated, then which case

A

the congruence

Theorem 7.8.

Part

(5)

can be ~

If

S

is

taken to be finite, in

is finitely

generated

by

of Theorem 7.5 then implies that

is generated by a finite set {X^i -

> whence

I

A

is

also generated by a finite set of pure difference binomials. Conversely, isomorphic to

any such residue class ring

R[X;F]/({Xgot -

X^a I ga »ha € F})

suitable subsets gruence on

F

F *

Let

generated by

Theorem 7.5 shows that and hence

R[{X^}]/A

ring over

R .

R[{X^}]/A for

P

t^16 con"

• Part

R[X;F]/({Xga -

is

(5.)

of

Xh a }) * R[X;F/p]

is, to within isomorphism,

,

a monoid

This completes the proof of Theorem 7.11. Section 7 Remarks

We have mentioned in this section that a general semi­ group ring

R[X;S]

of a monoid ring

can be defined as an appropriate ideal R*[X;S*]

advantages of working with regarded as an element of

over a unitary ring R1

and

R'[X;S’]

the definition of multiplication in can always t,u e S'

.

be expressed in the form

S'

Rf .

are that

X

Two can be

in this case and, in R f [X;S'] t*u

, each

s e Sf

for some

(In the proof in Theorem 7.1 that

(J) preserves

80

multiplication, s { S + S however;

for example, the possibility that

or that

t ^ T + T

has been ignored.

Not to fear,

the proof still works in those cases.)

The term unital extension of

R

has been used in the

paragraph following the proof of Theorem 7.1 without an ex­ plicit definition. S

of

R

The term means a unitary extension ring

such that

element of

S .

S = R + Ze , where

The papers

e

is the identity

[25] and [11] contain information

concerning the possible unital extensions of a given ring even in the case where

R

is already unitary.

Under what conditions is

R [ X ;S ]

conjecture would seem to be that only if

R

is unitary and

ditions on since that

R

R

and

S

S

unitary?

R [ X ;S ]

imply that

R [ X ;S ]

unitary implies

R

A reasonable

is unitary if and

is a monoid.

is a homomorphic image of

R[X;S]

Indeed, the con­ is unitary, and

R [ X ;S ] , it is also true

is unitary.

If

tains a cancellative element, it is also true that unitary implies that For example, let

S

R ,

S

con­ R [ X ;S ]

is a monoid, but not in general.

S = {a,b,c}

be the semigroup with the

following Cayley table. +

a

b

c

a

a

c

c

b

c

b

c

c

c

c

c

For any unitary ring

R , the element

X a + X^ -

identity element for

R[X;S]

is not a monoid.

, but

S

XC

more on this topic* see [74]. Theorems 5.10 and 7.8 show that a monoid

S

is

is an For

81 Noetherian if and only if semigroup,

S

is finitely generated.

For a

I do not know if either of these conditions im­

plies the other.

Similarly,

I know of no results beyond

Theorems 7.10 and 20. 7 concerning the problem of determining conditions under which a semigroup ring is Noetherian.

82 §8.

Zero Divisors

Two problems are considered in this section. is that free

of determining conditions under nontrivial zero divisors ----

of

an integral Theorem

which

8.1.

f e R[X;S]

R[X;S]

that is,

is

R[X;S]

is

domain. This problem is easily settled

in

In the second problem we take a fixed

element

and we seek necessary and sufficient conditions,

usually in terms of the coefficients of f

The first

should be a zero divisor in

R[X;S]

f , in order that .

The more definitive

results on the second problem require some type of restric­ tion on

S

or on

R.

Theorem 8.1. R[X;S]

R *

{0}.

Proof.

S

Assume first that

admits a total order ation.

Let

f,g

f = r i « l £ ix S l

R

f^

S1 + tl e

<

fg .

By Corollary 3.4,

S

compatible with the semigroup oper­

E i=.l8ix t l ’ w h e r e

and

R[X;S]

s l < - - ‘<s m

and write ’

g^ are nonzero. Then s I +11 ^l^i^ is the corresponding

and

in

is an inte­

is an integral domain and

be nonzero elements of

’ 8 =

t^<...
R

is torsion— free and cancellative.

is torsion— free and cancellative.

term

The semigroup ring

is an integral domain if and only if

gral domain and

S

Assume that

In particular,

fg *

0

so

R[X;S]

is an

integral domain. If

R

is not an integral domain, then choose nonzero

elements a,b e R s s aX • bX = 0 in domain. s,t,u £ S

Similarly, are such

so that R[X;S] if that

S

ab = 0 . , so

If

R[X;S]

s e S , then is not an integral

is not cancellative and if s +t = s + u

but

t * u

, then

83 for

r e R\{0}

we have

and

rX1” — rXu

rXs (rX*" -

are nonzero.

rXu ) = 0 , where

Finally, assume that

integral domain and that

S

sion— free.

be such that

for some

Let n e

ks = kt .

s,t e S

Z+ , and

S

is an

is cancellative, but not tor­

choose

k e Z+

s * t

while

ns = nt

minimal so that

=

(rxs -

rXt) ( S ^ J r X ^ ' 1'1^

is cancellative, the choice of -

(k -

R

r e R , r * 0 , then

If

0 = r2XkS - r2X kt Since

rXs

l)s + i^t * (k -

i2 -

k

implies that

l)s + i2t

0 < i^


Thus,

_k-l (k-i-l)s + it Si=orX * 0 , so again

+ Il:) .

for

1 .

R[X;S]

is not an

integral domain in this case. COROLLARY 8.2. Then

A [ X ;S ]

prime in

Assume that

is prime in

R ,

(2)

S

A

R [ X ;S ]

is a proper ideal of if and only if

is cancellative, and

(1)

(3)

S

A

R. is

is

torsion— free. Proof.

Corollary 8.2 follows from Theorem 8.1 and from

the isomorphism

R [ X ;S ]/A [ X ;S ] ~ (R/A)[X;S]

COROLLARY 8.3. cancellative and that that

fg = 0 . Proof.

fg e P[X;S] or

Let

P

S

f = Ef^XS* f j

is torsion— free and and

g = EgjX^

is nilpotent for all

be a prime ideal of

, which is a prime ideal.

g e P[X;S]

arbitrary,

Then

Assume that

, and in either case

it follows that

.

R .

is nilpotent.

i

and

j .

Then

Therefore 6 ^

are such

*

f e P[X;S] Since

P

is

84 If the semigroup

S

is ordered under a fixed total

order, then the usual notions of degree and order of elements of

R[X;S]

can be defined.

Thus,

if

f =

t*10

canonical form of the nonzero element

f e R[X;S] , where

sl < s2 < ••• < sn , then

the degree of

we write

deg f = sn .

andwe write fn = 1 . R[X;S]

ord f If R

sn

is called

Similarly,

s^

f

and

is the order of

= s^ ; the element

f

is monic if

is unitary, theset of monic

forms a multiplicative system in

f

elements of

R[X;S]

.

If

R

is

an integral domain, then we have, as usual, deg

(fg) = deg (f) + deg (g)

and

ord

(fg) = ord (f) + ord (g)

for f,g e

It must be understood that general,

degf

and

intrinsically determined by

depend upon the relation under which

R[X;S]\{0)

ord f

are not, in

f e R[X;S] S

.

; they

is totally ordered.

What can be said about zero divisors of

R[X;S]

if one

of the three conditions (1)

R

is an integral domain,

(2)

S

is cancellative,

(3)

S

is torsion— free,

is not satisfied? R[X;S]

can still be characterized --- at least in terms of

the structure of dition

It turns out that the zero divisors of

(2)

R --- if hypothesis

is the most crucial

(1)

of the three conditions in a

determination of the zero divisors of

R[X;S]

of polynomial rings, McCoy [104] proved that zero divisor if and only if ment

c e R .

is dropped; con­

cf = 0

.

In the case

f e R[X]

is a

for some nonzero ele­

This result extends to the case of semigroup

85 rings

R[X;S]

, where

S

is torsion— free and cancellative,

but in order to prove a bit more, we introduce the notion of an

S— graded ring. Let

that

R

R

be a ring and let

is

subgroup

w E _R is the weak seS s

group, of the family

ring

R

^Rs ^ses

s ,t e

^Rs ^ses

’’assume that

u „R_ Seo s

natural

S— grading of

R .

A given

S— grading for a given

^Rs ^S eS

an

’’assume that

R

is

R"

S— graded” . of

while elements of

are said to be nonhomogeneous.

S— grading of the semigroup ring that where we take

S ,

S— grading of

S— grading of R , then elements

are said to be homogeneous,

R\(us ^gRs )

namely,

an

abelian

S .

caH-ed an

is more descriptive than If

such that

direct sum, as an

may admit more than one

so the phrase

R

> and

R R c R for s t “ s+t

The family

We say

s e S , there exists a

of the additive group of

R =

(2)

be a semigroup.

S— graded if for each

Rg

(1)

S

There is a

A = R[X;S] ---

A g = {rXS | r c R}

for each

s e S . THEOREM 8.4. R , where

S

Assume that

{

geneous element of

f

homogeneous element and where for

If

f e R

R .

A nonzero element g n g = » where each

Rg

S-grading of

is annihilated by a nonzero homo­

Proof.

the same

is an

is torsionHiree and cancellative.

is a zero divisor, then

position

R c S S 6u

i * j .

g^

of

R

g^

is a nonzero

and

has a unique decom-

g^

do not belong to

To prove the result, we choose

86 a nonzero element

g e Ann(f)

whose number

n

geneous components is as small as possible. of the theorem amounts to saying that total order on Let

S

f = zm f i=l i positions of f

.

We consider f m &g

composition.

n —

.

have

1

Therefore

Consequently,

f.g = 0

If

for

each

^Rs ^ses

RsRu c R s+U

of

<j> : S

{R SRU

and since

follows that

f

and fg = (f +...+f ..)g = 1 m-1

choice of

This

g , we

completes the proof.

S— grading of R , then

of

t,v e T , then

abelian group by

annihilates

consequences of Theorem 8.4, we note

an

T— grading

if

f^g

fgn = 0 .By

At = £{RS I .

Moreover,

, * we obtain a

By induction it follows that

i , so

jective homomorphism

let

be a

components in its homogeneous d e ­

, and hence n = 1 .

induces a

fm mg &* 0

g , for

f g = 0 m

Before stating some

T

<

fm ^g^ = 0 , and the argument just given

fm ^g = 0 .

g = gR

that if

Let

and

and it has at most

shows that

n = 1 .

compatible with the semigroup operation.

contradiction to the choice of

0 .

The conclusion

g = E*1 g. be the homogeneous decomj= l j and g , respectively, where f. e R with 1 si and g. e Rt . with t., < t < ...
s. <...< s 1 m m &n = 0

of homo­

-► R,

T of

For

R = SteTA t

W

t

e T

generated as an t and

<|>(u) = v}

Hence

^t^teT

is a

.

Since

t + v , it T— grading

R . COROLLARY 8.5.

the semigroup cellative.

If

S

Let

A

such that f e R[X;S]

be the smallest congruence on S/A

,

clear*

<J>(s + u) = 4>(s) + <J>(u) =

A tAy c A t+y .

sur­

onto asemigroup

as follows.

That

I <J>(s)=

<J>

each

is torsion— free

is a zero divisor,

and can­

then there

Clearly

87 exists a nonzero

element

g e R[X;S]

suchthat

contained in an equivalence class of Proof. T = S/A

The

induces

A and

Supp(g)

fg =

canonical homomorphism <J>

of

a

Since

T— grading of

R[X;S] .

is

0 .

S

onto T

is tor­

sion— free and cancellative, Theorem 8.4 implies that

f

annihilated by a nonzero

But

T— homogeneity of Supp(g)

g

T —homogeneous element

g .

is

is equivalent to the statement that

is contained in an equivalence class of

A .

Inci-

j*

dentally, of

if

f , then

f = 2^=i^i fg = 0

COROLLARY 8.6.

t*ie

implies

T “ homogeneous decomposition

f^g = 0

for each

i .

Assume that S is torsion— free and cann sf = .X 1 is the canonical form of

cellative and that

f e R[X;S]

the nonzero element

.

The following conditions

are equivalent. (1)

f

(2)

There exists a nonzero element

rf^ = 0

is a zero divisor in

for each

Proof.

R[X;S]

. r

eR

such

that

i .

That

(2)

implies

(1)

is clear.

The converse

follows from Corollary 8.5, for that result implies that s there exists r e R\{0} and s e S such that 0 = rX • f = E^.^rf^Xs+si . Since rf^ = 0

for each

(1)

If

isprimary in (2)

If

follows

that

i .

COROLLARY 8.7. the unitary ring

S is cancellative, it

Assume that

A

is a proper ideal of

R .

A[X;S]

is a primary ideal of

R

and

A

is primary in

R [ X ;S ] , then

A

S is cancellative. R

and if

S

is torsion— free

88 and cancellative, then Moreover,

if

P = rad(A)

Proof. primary in

If R

cancellative, t * u .

A[X;S]

A[X;S] since

then

is primary in

, then

R[X;S]

.

P [ X ;S ] = rad(A[X;S])

.

is primary in

A = A[X;S] X s (X1 --

R[X;S]

n R .

Xu ) = 0

If

X s (X1 s and no power of X

is

the assumption that

A[X;S]

in

s e S

for some

- Xu ) e A[X;S]

Thus

, then

is

is not

t,u e S

, X1 -

A[X;S]

A

with

XU J A[X;S]

,

.This contradiction to

is primary shows that

S

is

cancellative. By passage to

R [ X ;S]/A[X;S]

consider the case where is a zero divisor in

in

A = (0) . v J

R[X;S]

(2)

Thus,

, it suffices to if

*

f = E11 -f.XSi i=l l

, then Corollary 8.6 implies

that there exists a nonzero element

r e R

rf^ = 0

is primary in

f^

for each

i

. Since

is nilpotent, and hence

proves

(2) .

(0) f

is also nilpotent.

It is clear that

This

P[X;S]

, and

is prime in

by Corollary 8.2.

In

(1)

torsion— free. istic

R , each

P[X;S] c rad(A[X;S])

the reverse inclusion follows since R[X;S]

such that

p * 0

of (8.7), the semigroup For example, and

G

if

K

S

need not be

is a field.of character­

is the cyclic group of order

pR ,

then the group ring K[G] is isomorphic to K[X]/(Xpn - 1) n ~ K[X]/ (X - l)p , a special principal ideal ring. Hence each ideal of

K[G]

is primary, but

The same example shows that tain

rad(A)[X;S]

if

S

G

rad(A[X;S])

is a torsion group. may properly con­

is not tors ion— free.

In order to continue our investigation of zero divisors of

R [ X ;S ] , we return to Corollary 8.5 and the notation used

89 there.

Thus,

nonzero and

assume that g

is

fg = 0 , where

T— homogeneneous, with

the congruence on S defined by + for some n e Zand some c e S . decomposition of ^g

= 0

f

f

for each

into i .

a A b If

and

g

T = S/A if

are and

A

na + c = nb + c n f. is the i= l i

f = E

T— homogeneous components, then

Let

4>*:R[X;S] -- ► R [X ;T ]

be the

homomorphism induced by the canonical mapping Then

<J>*

R[X;S]

maps each

onto

T— homogeneous element

<J>:S — *T a• Eh.X of

.

m

a monomial

(Eh )X ofR[X;T] . It is i s easy to show, however, that a monomial rX is a zero di ­ visor if and only if either s

is not cancellative in

r S .

is a zero divisor in Since

semigroup, we are able to conclude: ag = ’EgjX ^ , then (Zfij ) (Egj ) = 0 that

Ef^j

where

y

and

Egj

are

y(f^)

is a cancellative s .. if f .= Ef..X and l ii J for each i. We note y (g) , respectively, R[X;S]

integral domain, it follows that either y(f^) = 0

for each

i .

or

T

and

is the augmentation map on

R

.

If

y(g) = 0

In the case where

S

R

is an

or

is cancel­

lative, we prove a form of the converse in Theorem 8.9.

The

proof of (8.9) uses an auxiliary result. THEOREM If then

x e R x

8.8.

Let N

is such that is a zero

be the nilradical x

isa zero

of the ring

divisor modulo

R . N ,

divisor.

Proof. Choose y e R\N such that xy e N , say k k k x y = 0 . If xy = 0 , then x is a zero divisor. And if k i k xy * 0 , we choose i maximal so that x y * 0 . Since i k x • x y= 0 , it follows that x is a zero divisor. THEOREM 8.9.

Assume that

D

is an integral domain and

90 S

is cancellative.

such that

S/ a

Let

A

be the smallest congruence on

is torsion— free and cancellative and let

be the kernel ideal of the congruence of zero divisors in Proof. is unitary. onto

S/A

D[X;S/A]

D[X;S]

.

Then

I

I

consists

.

We prove theresult first in the case where

D

Denote by

S

and by

Since

D[X;S]

A .

S

Let

<J>*

<J>the canonical homomorphism of

the induced homomorphism on

is an integral domain, f e I .

I

The proof of part

D[X;S]

.

is prime in (2)

of Theorem m

7.2 shows that each n^

is expresible in the form s• t• is of the form r^(X 1 - X -1) with

f^

f

be the smallest positive integer such that

we assume that

n.

>1

is a zero divisor, of

X

s

-

X

for each

» where s^ A t^.

Let

n ^s i ~ ^ ± ^ 1

i .Before proving that

» f

we consider certain multiples in D[X;S]

t

= h , where s A t . If k is any positive ks kt k-1 (k-i -l)s+jt integer, then X - X = hg , where g =

We note that(k-j-l)s + jt A (k s A t . char D

Hence

=0

—

<J>*(g) = k X ^

or if

char D

and

=p * 0

in either of these two cases.

l)s

and

for each

j

<J>*(g) * 0 p f k ;

since

if thus, g ^

I

We return to the proof that

f

is a zero divisor, considering separately the cases where the characteristic of If 1

char D

D

is, or is not, zero.

= 0 , then the proof above shows that for

< i < m , there exists

f.g. = r. (Xn ^ * ii i g^...gm * 0 If

X

e D[X;S] \ I

= 0 .

Therefore

since this product, is not in

ch ar D

= p

n^ = m^p

such that fg . ..g = 0 l m

n.

, where

.

If n.

e^ > 0

is divisible by and

m^

and

I .

* 0 , then there are two cases

sider, depending upon write

g^

to con­ p , we

is relatively

91 prime to

p .

There exists

gj I I

such that

figi =

m isi m iti pe i r^ (X — X ) , a nilpotent element since (^igi) nei n is i n i^ i r? (X - X ) = 0 . If n^ is not divisible by p , then there exists is prime,

g. I I 6i T

f.g. = 0 . i6i g k I

it follows that there exists

fg = E^f^g

is nilpotent.

the nilradical of a zero divisor. D

such that

Hence

D[X;S]

f

Since

I

such that

is a zero divisor modulo

, and Theorem 8.8 shows that

f

is

This completes the proof in the case where

is unitary. If

D

is not unitary, let

the quotient field of element then

e .

If

I*

generated by

be the subring of

D

and its identity

is the kernel ideal of

I = I* n D[X;S]

g e D*[X;S]\{0} dg

D

D* = D[e]

.

Thus,

such that

if

on

D*[X;S] ,

f e I , there exists

fg = 0 .

is a nonzero element of

A

D[X;S]

If

d e D\{0> , then

that annihilates

f .

The proof of Theorem 8.9 abounds with examples of zero divisors in

D[X;S]

that need not be in

has characteristic

that is not in subsets of

S

I .

and

In fact, if

D

is not torsion— free, then k-1 (k-j-l)s+jt contains a zero divisor of the form 2j=ox

D [ X ;S ]

0

I .

S

Theorem 8.9 enables us to determine what

can be realized in the form

some zero divisor

f

of

D[X;S]

Supp(f)

for

, at least in the case where

Id| >2 . THEOREM 8.10. with

|D| > 2

that

A =

Assume that

D

is an integral domain

, that S is a cancellative semigroup, and n is an n—element subset of S . Then A

is the support of a zero divisor of

D[X;S]

each

with

a^

is

A— related to some

aj

if and only if j * i .

92 Proof.

The condition given is necessary, for suppose m , where f is a zero divisor. If f = ^ = 1

A = Supp(f) is the

T—homogeneous decomposition of

then Corollary 8.5 implies that each in

D[X;S]

.

f , where f^

T = S/A ,

is a zero divisor

Since a nonzero monomial is not a zero divisor,

it follows that each

a^ e A

is A — related to some

aj

with

j * i . To prove the converse, Theorem 8.10 shows that it is sufficient to prove the following statement: B = {b

...,b^}

is a

k— element subset of

and any two elements of support of an element g =

E^=i(dXb2i -

k = 2r + 1 |D | > 2

of

c + d * 0 b 2r+l

I .

If

where

cX l T

and let

k = 2r

G = (0,a,b,c}.

A = {0,a,b}

is the

is even, take

d e D\{0} .

g = Z. , (dX

|D | > 2

If

D

S

dX

D[X;G]

R[X;S]

is a group.

A—related elements of

+

of order

D[X;G]

G , but with support

2 . S , questions concerning

can sometimes be reduced to the Theorem 8.11 provides two tech­

niques for accomplishing such reductions. THEOREM 8.11.

D

is the field of two elements

For a cancellative semigroup zero divisors of

of

is the Klein four-group, then

is a set of

A , is a unit of

-

c,d

is necessary in the state­

X a + X b , the only element of

case where

B

k > 1

.

ment of Theorem 8.11, for if

f = X°+

A— related, then

Thus, choose nonzero elements

The assumption that

and

S , where

is odd, this is where the assumption that

such that (cX Lx -

g

are

d X b2i_1) ,

comes in.

b 2r

B

if

Assume that

S

is a cancellative

93 semigroup with quotient group (1)

f

is a zero divisor in

is a zero divisor in (2) of

S

R[X;G]

Assume that

containing

divisor of

S

0 .

R[X;S]

R[X;S]

.

if and only if

f

.

is a monoid and If

f e R[X;S]

G , and let

f e R[X;H]

if and only if

H

is a subgroup

, then

f

f

is a zero

is a zero divisor of

R [X;H] . Proof,

is a unital extension of R , then s is a subring of R*[X;G] and X is a unit of

R[X;G] R*[X;G]

If

R*

for each

se S .

for

b e R*[X;G] .

and

(2) .

Thus,

zero element of XSh e R[X;S] that in

f (1)

Hence

X Sb =

implies

b = 0

We use this fact inproving both

assume that R[X;G]

, and

fh = 0 ,

. There exists

XSh * 0 .

is a zero divisor in

where

h

s e S

(1) is

a no n­

such

that

Since

fXSh = 0 , it follows

R[X;S]

.

The other assertion

is patent.

Similarly,

f

a zero divisor in

a zero divisor in

R [ X ;S ] .

nonzero element of

R[X;S]

induces a grading of

R[X;G]

S = G .The ; let

ofb .

T — homogeneous decomposition

of

for each

for some (1) ,

i.

b* e R[X;H]

If

f

b

is

be a

such that fb = 0 ; without loss

T — homogeneous decomposition

fb^ = 0

R[X;H]implies

For the converse, let

of generality, we assume that

in

0

b = Since

0 = fb is

s e SuppCb-^) ,

. Therefore

group

T =

G/H

be the f e R[X;H] n ^1 = 1 ^ ^ . then

b^

0 = fb^ = fX b*

, the Hence = XSb*

, and as

fb* = 0 .

The proof of Theorem 8.11 establishes the following corollary.

94 COROLLARY 8.12. monoid and

H

f e R[X;H]

Assume that

is a subgroup of

S S

is a cancellative containing

, then the annihilator of

f

0 . If

in R[X;S]

rated as an ideal by the annihilator of

f

in

is

R[X;H]

gene­ .

As an application of Corollary 8.12, we determine cons ditions under which 1 - X is a zero divisor, as well as the annihilator of this element. THEOREM 8.13.

Assume that

is a cancellative monoid.

R

is unitary and s f = 1 - X is

The element

S a

zero divisor in R[X;S] if and only if ns = 0 for some + n e z . If n is minimal so that ns = 0 , then g = 1 +

X S+ ...+ X Proof.

generates the annihilator If f

shows that each divisor. 0 A s —

Since

of

is a zero divisor, then Corollary

A—homogeneous component of 1

that is,

f .

is not a zero divisor, ns = 0

f

it

is a zero follows that

n e z+ .

for some

8.5

On the other

hand, the proof of Theorem 8.1 (or the next paragraph) shows that

f If

is a zero divisor if n

s

is minimal so that

subgroup of

S

of order

and

0

are

ns = 0 , then

n , and

H

A—related. <s> = H

contains

0 .

Corollary 8.12, it therefore suffices to prove that rates the annihilator of fg = 0

is clear.

And if

f

in the case where (1 -

is a

By g

gene­

S = H .

That

X S) (aQ + a-|XS+ . . .+an _ ^ n 1^S)=

0 , then a calculation of the coefficients in the product leads to

the system of equations

. . . = a^ , n-l is Ea^X = a^g

sl

0

n- Z

.

Therefore

0 =aQ -

a n _i ~ a i ~

a0

a n = a, = ... = a„ and 0 1 n -1

is in the ideal generated by

g .

=

95 COROLLARY 8.14. R

If

is a unitary ring,

for each

r

G =

is a finite group and

kg* h= E-^X51

then

annihilates 1 -

g

X

g G G .

Proof.

Let m

n .

If

of

(g) in

is a complete set of coset representatives k a- n-1 ig G , then h = (E^= ^X ) (E^ = 1X ) . Thus, Theorem

8.13 shows that

f = 1 —

Xg

andassume that

h annihilates

g

has order

f .

Corollary 8.14 yields an extension of Theorem 8.9 to the case of a coefficient ring with zero divisors. THEOREM 8.15.

Assume

cancellative semigroup. such that

is

a zero divisor of R[X;S] Let

ls-1^

for each

of

for

S .

^i^i=l

^ri^i

extension of If g^ = t^ Hof

* - X

R s^

G .

e

Let

of

R •

and let G ,

G

then

H

=

khf and ^et

annihilates each r e R\(0)

f = E^ r ^(X

each subset

generates a finitesubgroup

1 k

is a ring and S is a ,,n u is a subset of

i , then

R* be a unital

be the quotient group {g*}?

R

-1n

r

If

S

Proof.

s^ A t^

that

^

= S1X ‘ Corollary 8.14 shows that g* 1 — X 1 , and hence hf = 0 as well.

and if

nonzero annihilator

s = E^s^ , it follows that of f

in

rXSh

h If

is a

R[X;S]

.

Section 8 Remarks The reader has probably noted the crucial role played by the assumption that

S is

cancellative in this section.

fact, Corollary 8.5

is essentially the only result about

In

zero divisors in the section

that does not use the cancella­

tive hypothesis on

topic of zero divisors of

S .

The

96 R [ X ;S ] , in the case where

S

is not cancellative,

is

largely unexplored territory. Theorems 8.9 and 8.10 stem from the doctoral disserta­ tion of Janeway

[81].

J an eway’s thesis contains additional

results on zero divisors that have not been included here. Another result that we have not included is a form of the Dedekind-Mertens Lemma for semigroup rings.

For poly­

nomial rings, the Dedekind-Mertens Lemma states that if f»g € R [- C } ] , then there exists a positive integer k k+1 k such that c(f) c (g) = c(f) c(fg) . This lemma can be used to give an alternate proof of McCoy's Theorem on zero divisors in

R[(X^}]

.

For a torsion^-free cancellative semi­

group

S , it's true that if f,g e R[X;S] \{0> and if k k-1 k = |Supp (g) | , then c(f) c(g) = c(f) c(fg) . The proof hasn't been included in Section 8 because of limited appli­ cability of the result.

For additional information on the

Dedekind-Mertens Lemma and for its proof, the interested reader may consult

[111],

[51, Section 28], or [54].

97 §9.

Nilpotent Elements

In considering nilpotent elements of again two basic problems.

R[X;S]

, there are

One is the global problem of de­

termining conditions under which

R[X;S]

is reduced.

This

is closely related to the problem of determining conditions under which

N[X;S]

is the nilradical of

since

N[X;S]

R[X;S]

, it follows that

only if

R[X;S]

is clearly contained in the nilradical of

(R/N)[X;S]

N[X;S]

is the nilradical if and

is reduced.

The local problem is that

of determining,

for a given element

under which

is nilpotent.

f

set for the nilradical of

f = Er^X

si

, conditions

One of our main approaches to

the local problem is to attempt

to determine a generating

R[X;S]

.

Our first result indi­

cates three ways in which nilpotent elements of arise.

, for

Recall from Section 4 that elements

R[X;S]

a,b e S

are

said to be asymptotically equivalent if there exists such that

na = nb

for each

p— equivalent for a prime THEOREM 9.1. N

and that

Each

(2)

If

such that

pr

(3) a rX

If

R

a

K € Z+

and

b

for some

are k ^ 0 .

is a ring with nilradical

is a semigroup. element of

N[X;S] is nilpotent.

a,b e s

are

a,b e S

are asymptotically equivalent, then

p— equivalent and if r e R is a b is nilpotent, then rX — rX is nilpotent.

b -

rX

is

nilpotent for each r e R .

Proof.Statement (3)

k k p a = p b

if

Assume that

S

(1)

p

n > K , while

may

, thereis no loss

unitary since

R[X;S]

(1)

is clear.

In proving

(2)

of generality in assuming that is a subring of

R*[X;S]

,

and R

is

where

R*

98 is a unital extension of

R .

In

(2)

k k p a = p b ,then

, if

(Xa - X b )p k e pR[X;S] , and hence [r[Xa - X b )]p k c p r pkR[X;S] , a a b nilpotent ideal. Therefore r(X - X ) is nilpotent. To prove (3)

, we show that

X a - Xb is

such that

na = nb

for each

show that

(Xa - X^)m

integers such that v £ K , and hence ua

n

> K .

= 0 . If

be

u

If

m = 2K + 1 ,

we

and

v are nonnegative

u+ v = m ,then either ua = ub

+ vb = (u + v)a = ma . (Xa -

K e Z

nilpotent. Let

or

vb = va .

u >

K or

In either case,

Therefore

X b )m = E ; . 0( -l) 1( ? ) X (m'l)a+ib =

x mai™=0( - D x (” ) = x ma(i -

l)m = 0 .

Several necessary conditions in order that

R[X;S]

should be reduced follow from Theorem 9.1, but rather than state them here, we allow them to appear in various theorems as conditions that are both necessary and sufficient in order that

R[X;S]

should be reduced.

There is a natural dichotomy

into the cases of nonzero characteristic and characteristic 0

in considering the nilradical of

the case where char R THEOREM 9.2. acteristic

p

that is,

R

Assume that

R[X;S]

.

We begin with

* 0 . R

and with nilradical

nilradical of

R[X;S]

if and only

is a nil ring, or

(2)

is a ring of prime N.

Then

N[X;S]

if either (1) R * N

and

char­ is the

R = N — S

is

p— torsion free. Proof. that

(1)

radical of

In view of part

or

(2)

R[X;S]

(2)

is satisfied if .

of Theorem 9.1, it N[X;S]

For the converse,

follows

is the nil-

it suffices, by

99 passage to

(R/N)[X;S]

, to prove that

R[X;S]

is reduced if

R

is reduced and S is p— tors ion— free. Thus, let sA f = If^X be the canonical form of the nonzero element

For any positive integer Pk d V Ef^ XF 1 since

s^

Therefore R [ X ;S ]

; and •pk f^

s-

are not

p— equivalent for

* 0 for each

k , and

9.3.

is

IfI>

each

i

i * j

.

is and

this implies that

p * 0 ,

is a nonzero integral domain of then

D[X;S] is reduced if and only

p— torsion— free.

As in Section 4, we use

g

9.4 to denote the congruence of group

nk f^

is reduced.

characteristic S

Pk f^ * 0 for

this is true since

COROLLARY

if

k , the canonical form of

f .

in the statement of Theorem p— equivalence on the semi­

S . THEOREM 9.4.

acteristic

pn .

Let

R

be a ring of prime-power char­

The nilradical of

N [ X ;S ] + I , where

N

is the nilradical of

the kernel ideal of the congruence Proof.

R[X;S]

Theorem 9.1

is the ideal R

and

I

is

~ .

shows that

N[X;S]

is con­

tained in the nilradical of

R[X;S]

is clear if

R * N , we establish the reverse

R = N , and if

inclusion by showing that

.

+ I

The reverse inclusion

R[X;S]/(N[X;S] + 1 )

is reduced.

By Corollary 7.3, this residue class ring is isomorphic to (R/N)[X;S/p] and hence

.

R/N

Since

R

has characteristic

has characteristic

p .

pn , pR c n

Because

S/g

is

p— torsion— free by Theorem 4.5, it follows from Theorem 9.2 that

(R/N)[X;S/p]

is reduced.

This completes the proof.

,

100 If if

G

is an abelian group of finite exponent

el ek n = p^ •••Pfc

is well

the prime decomposition of

known that

G

decomposes as a direct r . e• . G- - Ig e G | Pj^g = 0^ •

G * G ^ Q - . ^ G ^ , where eA G^ has exponent p^ ponent of

G .

If

n

n , then it sum The group

and is called thep^— primary

G

and

com­

is, in fact, a ring considered as a

group under addition, then each

G^

is an ideal of

G .

We

use these facts in the statement of Theorem 9.5. THEOREM 9.5.

Assume that R is a ring of nonzero charei ev n = Pi ••-P^ > and let R = R^S.-.eR^ be the

acteristic

decomposition of R e, char R i * Pi • If

into primary components, where N

is the nilradical of

N[X;S] + £^= 1 ({rXa “ nilradical of Proof. composition radical of

R , then

I r e R^

and

a ~b})

is the

R [ X ;S ] . The decomposition R[X;S] = ^© R^ fXjS]

R[X;S]

the components

R = of

induces the deR[X;S]

, and the nil-

is the direct sum of the nilradicals of

R^[X;S]

.

If

is the nilradical of

R^ ,

then Theorem 9.4 shows that N^[X;S] + ({rXa — of

rX^ | r e R^ , a ~

R± [X; S] ; here

(

)

b})

is the nilradical

is used for ’’ideal of

R± [X;S]

generated by the given set11, but this is the same as the ideal of ical of

R[X;S] R [ X ;S ]

generated by the set.

Thus,

the nilrad­

is

(N.[X;S] + ({rXa - rX^ I r e R. and a ~ b})) = i= l i , , 1 Pi N[X;S] + E* , C(rXa - rX | r £ R. and a ~ b}) . 1 1 Pi COROLLARY 9.6. Theorem 9.5.

Then

Let the notation and hypothesis be as in N[X;S]

is the nilradical of

R[X;S]

if

101 and only if

S

is

p^— torsion free for each

is not a nil ring; is reduced and

S

R[X;S]

is

i

such that

is reduced if and only if

p^— tors ion— free for each

R

i .

The results leading up to Theorem 9.5 provide an effec­ tive device for determining whether a given nilpotent,

in the case where

is as follows.

charR

The decomposition

respect to the decomposition

f^

is nilpotent.

case where

R

then

f

ajXSj

f -

such that

g

f = E^^ajX^

Sj

for

~ Sj

for

f - g -

(a^X

s^

and only if

f

j

the sum

f

First,

of all If

g = f ,

is nilpotent if and

.

We examine

>

s^ .

1 , then >1, then

-

g

.

Assume that the notation is

j sl

with

can

pe

is nilpotent.

Otherwise,

is nilpotent. n s-; g = Sj=i ajX J

f -

If

of

it suffices to consider the

such that a^

p— equivalent tono potent.

f =

is nilpotent if and only if

Thus,

is nilpotent.

only if

f

is

The procedure

has prime power characteristic

we can subtract from monomials

= n * 0 .

R[X;S] = E^_^@R^[X;S]

be effectively determined; each

f e R[X;S]

si a^X

f

f

If it is

is not nil-

is nilpotent if

is nilpotent, and

the latter element has its support of smaller cardinality than

f — g

does.

Continuing this process, we either settle

the question of nilpotency of (n - 1)sjt_

f - g

a composition

Alternately, everything

Supp(g^)

i , while elements of

to elements of

can be

bXs

to

handled

stage, as follows. The congruence g induces u f — g = ^i*iSi °f f ~ g into components

such that all elements of each

before we reach the

step or, at this step we reach a monomial

test for nilpotency. at the

f

Supp(gj)

for

are

Supp(g^) j * i .

p-equivalent for

are not

p— equivalent

102 the

T— homogeneous decomposition of

8, where R[X;S]

T = S/~ .)

If

y

f

considered in Section

denotes the augmentation map on

, then an examination of the proofs of Theorem 9.2 and

9.4 shows that for each

f

is nilpotent if and only if

uCg^) e N

i .

We turn to a consideration of nilpotent elements of R[X;S]

in the case where

ch a r R

= 0 .

be reduced to

the unital case,

tension of

, then the nilradical of

R

contraction to

R[X;S]

f = Ei=ia^XSi , then

Most questions can

for if R*

is a unital

R[X;S]

of the nilradical of f e R Q [X;S]

, where

is

integers.

what follows, we need to use some of

of

Z[ai,...,an ]

FMR— ring.

.

If

Rq - Z[a^,...,an ]

generated ring extension of the ring

perties of such rings

the

R*[X;S]

is a finitely In

ex­

Z

of

the pro­

Z[a^,...,an ] . The required properties

follow from the fact that it is a Hilbert

We therefore interrupt our treatment of nilpotent

elements in order to present a basic theory of such rings. We stick with the statement made in the preface to the effect that the reader is expected to be familiar with the results of [83].

Thus, Hilbert rings are treated in Section 1— 3 and

elsewhere in [83] and we do not repeat proofs of results that appear there. To recall the definition, ring

R

a Hilbert ring is a unitary

such that each proper prime ideal of

intersection of maximal ideals of

R .

R

is an

It is clear from this

definition that the class of Hilbert rings is closed under taking homomorphic images.

One of the basic results in the

theory is that the class is also closed under (finite) poly­ nomial ring extension; hence, each finitely generated ring

103 extension

of a

Hilbert ring is again a Hilbert ring,

this applies to Hilbert ring.

the rings

Z[a^,...,an ]

since Z

and

is

a

The other properties of the class of Hilbert

rings we need are contained in following theorem.

THEOREM 9.7. a

The following conditions are equivalent in

unitary ring R . (1)

R

(2)

R [X ]

(3)

For each maximal ideal

traction

is a Hilbert

ring.

is a Hilbert ring.

M n R

of

M

The unitary ring

to R

R

M

M

R[X]

is said to be an

of

, the con­

is maximal in

finite maximal residue class rings) each maximal ideal

of

R.

if

FMR— ring (FMR

R/M

The rings

R . for

is finite for

Z[a^,...,a ]

have

this property by the next result. THEOREM 9.8. Then each finitely also a Hilbert Proof. n

=1 .

R[Y]

Assume that

By induction,

it suffices to

= M n R .

finite, and

of

R

is

R[y^]

resolve the case

isa homomorphic

and since homomorphic images of Hilbert

Let

FMR— ring.

FMR— ring.

Moreover, since

FMR—ring. q

is a Hilbert

generated extension R[y^,...y ]

these two properties,

M

R

M

it suffices to show that

be a maximal ideal of

Then M / M q [Y]

M

q

is maximal in is maximal in

Since each maximal ideal of

F[Y]

image

of

FMR— rings have R[Y]

is an

R[Y] and let

R , F = R/Mq

is

R[Y]/M q [Y] ~ F[Y]

.

is generated by an irre­

ducible polynomial, the associated residue field is finite.

104 Hence

R[Y]/M

is finite, as we wished to show.

With Theorems

9.7 and 9.8 in

question of nilpotents of THEOREM 9.9.

tow, we return

R[X;S] .

If Sis tors ion—free, then

the nilradical of

to the

R[X;S]

, where N

N[X;S]

is

is the nilradical

of

R . Proof.

If

R

has nonzero characteristic, the result

follows from Corollary 9.6. R*

Assume that

char R

= 0 .

If

is the ring obtained by canonically adjoining an identity

element to

R , then

N

is the nilradical of

R* , so it

suffices to prove (9.9) under the assumption that R is n Si unitary. Assume that f = E^=1f^X e R[X;S ] is nilpotent. f e RQ [X;S]

Then

FMR— ring.

, where

The nilradical

maximal ideals since nM^

.

hence

RQ = Z [f ^ ,...,f ]

For each

Nq

RQ

of

Rq

is

f 6 M^[X;S]

say

Nq =

has nonzero characteristic, and

(Rq/M-^)[X;S] ~ R q [ X ;S]/M^ [X;S]

Therefore

an intersection of

is a Hilbert ring —

X , Rq/M^

is a Hilbert

for each

is a reduced ring.

X , and

f € n(Mx [X;S]) = N Q [X;S] c N[X;S]

consequently,

.

The following ancillary result is used in the proof of Theorem 9.11. THEOREM 9.10. characteristic ideals of

D

0

Assume that and that

such that

D

is an integral domain of

^Px^XeA

nP^ = (0) . A

finite set of prime integers and let of D/P^

A

consisting of all

X

is distinct from each

a Let Aq

prime 1 X— X

be a

be the subset

such that the characteristic of p^ .

Then

"X c A q ^X = (0) •

105 Proof. each

Let

a

e nA€A ^A * Then p,p0 *-*p a € P for 0 1 Z n A A e A , and hence p^...p a = 0 . This implies that

a = 0 , for if not, the characteristic of divisor of

would be a

PiP2 ***Pn *

THEOREM 9.11. characteristic S

D

Assume that

0 .

Then

D

D[X;S]

is an integral domain of is reduced if and only if

is free of asymptotic torsion. Proof.

torsion if

Theorem 9.1 shows that D[X;S]

is reduced.

is free of asymptotic torsion.

S

is free of asymptotic

Conversely, assume that S n si If f = £^= 1 f^X » then as

in the proof of Theorem 9.9, f e D q [X;S] FMR— ring

of characteristic

D q

0 .

for a Hilbert

Thus, to prove that

f = 0 , there is no less of generality in assuming that is a Hilbert

FMR— ring.

distinct elements of

We observe that if

S , then

s ~ t

many primes

p

such that

s^ ~ s^

the subset of

A

acteristic of

D/M^

D .

s

and

for some

t

i,j

{pt}^=1

(0) = nXeAMA » w ^ere

of maximal ideals of

t

are

are not

Thus, there are only finitely

1 ^ i < j < n ; we label this set as a Hilbert ring,

and

for at most one prime

p ; this follows from Theorem 4.6 since asymptotically equivalent.

s

D

.

with Since

CMx^AeA

D

family

Theorem 9.10 shows that if

consisting of all

A

is

A

0 such that the char­

is distinct from each

is

p t , then

CO) = n-v a M. . To complete the proof, we show that A 0 f e M^[X;S] for each A £ A q ; this is sufficient since n

M [X;S] = (0) . AgAq A

and that

D/M

y:D — >■ D/M

Thus, assume that

has characteristic

M

m

q ^ ^Pt^l

be the canonical homomorphism,

is maximal in *

D

Let

and let

y*

be

106 the canonical extension of Qk nilpotent, [v*(f)]4 By choice of

q

f e M[X;S]

i .

to

D[X;S]

.

Since

y*(f)

is

0 for some k e Z+ . m k ( p j , the elements q s^ are

and the set

distinct for each and

y

Therefore

yff^) = 0

as we wished to show.

for each

i

This completes the

proof of Theorem 9.11. COROLLARY 9.12. acteristic

0 .

Assume that

D

The nilradical of

is a domain of char­ D[X;S]

is the kernel

ideal of the congruence of asymptotic equivalence on A semigroup 2x = x + y = 2y

S

S .

is said to be separative if

implies

x = y

for

x,y e S .

The class of

separative semigroups is well known in semigroup theory.

The

next result shows that separative semigroups are the same as the semigroups that are free of asymptotic torsion. THEOREM 9.13.

Let

S

be a semigroup.

separative if and only if In particular,

S

Then

S

is

is free of asymptotic torsion.

a cancellative semigroup is free of asymptotic

torsion. Proof. x,y e S

If

S

is

not separative, then there exist

such that

2x

= x + y = 2y

and

3x = 2x + y = 2y + y =

3y , and hence x

totically equivalent.

This shows that S

S

x * y . Then and y

are asymp­

is separative

if

is free of asymptotic torsion. Conversely,

if there existx,y € s

x *

y , then choose k > 1

all

n ^ k .

z =

(k —

Then

minimal

2 (k — l)x =

l)x + (k — l)y .

If

with

such that

2 (k - l)y .

x ~ y but nx = ny

for

Let

z = 2 (k — l)x , then

S

is

107 not separative. S

On the other hand,

still fails to be separative,

2(k - l)y = 4 (k — l)x

and

(k - l)y = 4(k - 1)y =

if

for

z * 2(k - l)x , then 2z = 2 (k - l)x +

z + 2 fk - l)x = 3 (k -

l)x +

4 (k - l)x .

It is clear that a cancellative semigroup is separative. COROLLARY 9.14.

If

domain of characteristic

S

is cancellative and

0 , then

D[X;S]

D

is a

is a reduced

ring. In view of Theorem 9.5 and Corollary 9.12, a description of the nilradical of

R[X;S]

has characteristic

is missing only in the case

where

R

0

main.

Theorem 9.16 fills this gap without explicit mention

either of the characteristic of

and is not an integral do­

R

or of any hypothesis

concerning absence of zero divisors in

R .

The strategy of

the proof of Theorem 9.16 is easy to describe.

Let

be the set of proper prime ideals of

R

and let

p^ =

char (R/P.) for each A . Let I_ A PX the congruence £ on S , where ~

be

the kernel ideal of

denotes asymptotic

° equivalence on that

S .

It follows from Theorems 9.4 and 9.11

R[X;S]/(P^[X;S] + Ip ^)

Hence, the nilradical of Q(P^[X;S] + Ip^) element

f

•

R[X;S]

is contained in

of this intersection is nilpotent. is that

plies that some fixed power .

X

Theorem 9.16 shows that, conversely, each

needs to show, of course,

P^[X;S]

is a reduced ring for each

fn

What one

f e X nfP^tXjS] + In a P x) of

f

im-

belongs each

Theorem 9.15 treats two special cases of this

problem. THEOREM 9.15.

Assume that

P

is a prime ideal of

R

.

108 such that

R/P

has characteristic

kernel ideal of the congruence m s f = Ei=i£ ix

(1) to

Sj

p Sj

f e P [X;S ]+ I and if

for

i * j , then

s i>s j € Supp(f)

the relation

f

Proof.

Since each

,

(2) onto

Pk

I

be the

.Let

is

.

k

is such that

that are

e P[X;S] + I

notp-equivalent

implies that

maps to E ^ f ^ X ^ ^

*

[s.l

Let

y

(R/P)[X;S]

, which

f

Pk

R[X;S]

that

and

ideal of

By choice of

in(R/P)[X;S]

.

satisfies the hypothesis of

a ~ b })

, the V p ,

(1) , where the ideal

.

kernel k [y(f)]P P

in (1) im.

Let the notation be as in the paragraph

preceding the statement of Theorem 9.15. R [X :S ]

onto

f. e P for 1

is replaced by the zero ideal of R/P . Therefore (1) k nk plies that y(f^ ) = 0 , which means that fp e P[X;S] THEOREM 9.16.

.

be thecanonical homomorphism of R[X;S] k k . Then y(fP ) = [y(f)]P belongs to

y(I) = ({r*Xa - r*X^ | r*e R/P ~

e P[X;S]

isinP[X;S/~]

for i * j , it follows

J

p s^ =

p— equivalent, then

(1): Under the canonical map from f

[s.] 1 i .

S .

s^

f e P[X;S]

If the positive integer

for any

R[X;S/~]

on

and let

1 e R [x;s]-

If

(2)

~ P

p > 0

The nilradical of

is n

Proof.

A eA

{P [X;S] + I x

p

} . A

We have previously observed that the nilradical

is contained in the intersection described. observe that since nilradical of

N[X;S] + 1^

R[X;S]

For the converse,

is contained both in the

and in each

P^[X;S]

+ Ip^ , we can

109 pass to the residue class ring

(R/N)[X;S/~]

out loss of generality we assume that

R

.

Thus, with­

is reduced and that

S

is free of asymptotic torsion. We take an element m s , f = = in[X;S] + Ip } . As in the proof of X Theorem 9.11, it follows that the set A of primes p that say

s^ ~ Sj

for some

A =

with

1 < i < j < m

is finite,

(we presently address the case where

empty).

Part

for each

X

(1)

of Theorem 9.15 shows that

such that

f e n^P^[X;S] = (0) we choose

k

p^ ^ A .

and

f

Thus,

k

p s^ = p sj .

p^

e A.

Part

f t f

e P^[X;S] . If t ^ € Q P^[X;S] = f0) .

if

Take a prime

(2) of Theorem 9.15 = sup{pi >i=1

is

f e P^[X;S]

If

A * <J> , then

si ~ sj » with ^ ideal

A

A = <J> , then

is nilpotent.

large enough so that

k

implies

i,j

such

P^

p e A ,

such that

implies that , it then follows

that

This completes the proof.

Theorem 9.16 generalizes the three previous results C9.4),

(9.5), and C9.12) —

the nilradical of

R [ X ;S ]

of this section characterizing in special cases.

The way to

obtain a given special case from C9.16) may not be clear, however.

It is of some help to have alternate descriptions

of the intersection follows from part and for

P(P->[X;S] + X X (1)

i > 0 , let

Aj * {X e A | p x = p A ) .

In ). One such description Px

of Theorem 7.4. p^

be the

If

Cp^ =

Thus,

ithprime.

let

Define

> then Part

C7.4) shows that

" x ^ ^ s ] and hence

+

p^ = 0

= cP i [X:S] + TPi

’

C1)

of

110 V

a

CPa ^

s

] ♦ I

) A

CP,[X;S]

+ I

) = Px

oo n• i=oCcP i [X;S]

+ ip .) .

A third description of the nilradical — (9.12) as a consequence of (9.16) — nation the proof of (9.16).

the one that yields

follows from an exami­

To wit, if

defined in (9.16) and if

f

and

A

are as

| p^ | Al , then the

proof of (9.16), together with

(1) of Theorem 7.4, show that

f e (Ba [X;S] + IQ) n {^n^(Cp[X;S]

+ Ip ) } .

It is clear that

each such set is contained in the nilradical, and hence a third description of the nilradical is Z{(B.[X;S] A

+ I ) n [ n (C [X;S] u pcA P

+I_)]} P

where the sum is taken over all finite subsets

A

of the set

of positive primes. The description of the nilradical in (9.16) is strong enough to enable us to determine conditions under which R [X ;S ] of

R[X;S]

of new on the R

is reduced, and under which .

R

if nx = 0

n

implies

is said to be x = 0

for

is unitary, this amounts to saying that

element of

isthe nilradical

To state these results, we introduce

terminology. The integer ring

N [ X ;S ]

n

R , and in general it means that

THEOREM 9.17. and only if

(1)

totic torsion, and

The R

(3)

p

(2)

S

is regular on

e R ; if

is not element of

semigroup ring R[X;S]

is reduced,

regular

is a regular n

divisible by the additive order of a nonzero

x

one item

is R

R .

is reduced if free of asymp­ for each prime

m p

such that

S

Proof.

R[X;S]

conditions

(1)— (3)

p such that

empty,

then

R[X;S]

S

shows that conditions

is reduced.

the characteristic

square— free integer.

c

Let

R

element

of characteristic

show that

that

R

duced,

R* .

R

Let

each

V

generated by

is not

p

p

R , and hence

W .

To show

0

is re­

or

a ex­

= (0) .

W

is

S

is

of

R

In order that

p

R[X;S]

(0)

since p^

p^— torsion— free —

N[X;S]

Thus,

for each

Ip^ =

I = (0) PX p^ * 0 , then

N

is re­

such

for the nilradical of

then

Let

R[X;S]

In this case we examine the

This completes the

COROLLARY 9.18.

that

R.y[X;S]is reduced.

isa unit

Therefore, the nilradical of

R .

R

be the regular multiplicative

n(P^[X;S] + 1^ )

= 0 , A asymptotic torsion. If

N[X;S]

Since

be theunital

in Theorem 9.16.We observe that X . If

shows that

is either Ze

is

It is straightforward to

p— torsion— free.

representation R[X;S]

c .

it suffices to show that

S

R

W

Thus, we assume without loss of generality

we further assume that that

of

If

is reduced and that each element of

is unitary.

system in

be the set of

obtained by canonically adjoining an identity

R*

regular in

W

W * .

R* = R ©

tension of e

Let

is not p— torsion— free.

Assume that

are

Conversely, assume that

are satisfied. S

(1)— (3)

is torsion— free and Theorem 9.9

is reduced.

duced,

p— tors ion— free.

Theorem 9.1

satisfied if

primes

is not

is

S

for

is free of

is a nonunit of that is,

Ip = (0) .

n P [X;S] = A

A

proof.

be the nilradical of the ring

should be the nilradical of

R t X ;S ] , the following two conditions are necessary and

112 sufficient.

S

(1)

S

is free of asymptotic torsion.

(2)

p

is regular on

is not

R/N

for each prime

p

such that

p— tors ion— free. Section 9 Remarks

Section 31 of [51] is essentially self-contained,

and it

represents another source for the basic theory of Hilbert rings.

For additional information on separative semigroups

and some reasons they’re studied in semigroup theory, see Section 4.3 of [32] or Section 4 of [74]. The proofs of both Theorem 9.9 and Theorem 9.17 refer to the canonical adjunction of an identity element to a ring

R .

Besides the sources mentioned in Section 7 remarks concerning unital extensions of a ring

R , [51, Section 1] also con­

tains information on the adjunction of an identity element to a ring. For noncommutative rings

R , the term semiprime ring

is used instead of reduced ring.

Connell in [36, Theorem 5]

proves the following result.

R

ring and ring n

G

If

is a commutative unitary

is a (possibly nonabelian)

group, then the group

R [G ]

is semiprime if and only if R is semiprime and + is regular in R for each n e Z that is the order of

an element of

G .

Alternate versions of Theorem 9.17, using some of the properties of separative semigroups, may be found in [132], [129], and [30]. Much of the material in Section 9 first appeared in [117].

113 §10.

Idempotents

Given a unitary ring idempotent elements potents —

that

of R

is,

R , the two extremes in terms of are that

Rconsists of

R is a Boolean ring —

are the only idempotents of

R .

or

idem­ 0

and

1

In the second case,

said to be indecomposable, for in this case

R

R

is

is not ex­

pressible as a nontrivial internal direct sum of ideals.

In

Section 17 we examine the problem of determining conditions under which

R[X;S]

is Boolean or, more generally,

in the sense of von Neumann.

regular

We concentrate here on de­

termining conditions under which

R[X;S]

is indecomposable.

This is a special case of the problem of determining con­ ditions under which the idempotents of

R[X;S]

are in

R ,

and this more general problem is essentially no more diffi­ cult to resolve.

If

f = E^=1fiX Si e R [ X ;S ] , then we

determine several necessary conditions on in order that

f

should be idempotent.

Supp(f)

and

cff)

On the other hand,

Theorem 10.1 is one of the few results that give sufficient conditions for

f

to be idempotent.

THEOREM 10.1.

Assume that

R

is a unitary ring and

S

is an additive semigroup. (1) only if (2) n = |H|

A monomial r

and

s

H

=

If

considered in

(1) (1)

of

R [X ;S ]

is idempotent if and

are idempotent. is a finitesubgroup of

is a unit of

Proof.

group of

rXS

R , then

is obvious.

n

Z^X i

S

and if

is idempotent.

We note, however, that

r

is

as an element of the multiplicative semi­

R , rather than the additive group of

R .

In

114 (2) , let

n h. = Z1X 1 .

f

follows that

f2 = nf

hSince X Jf = f

and

for each

(n ^f)^ = n

We remark that each idempotent of and only if each idempotent of

is in

1 - f

contains an idempotent

f ^ R,

if

That

And if

then either

f

or

is anidempotent that is not a monomial, n s• If f = E^-^f^X is idempotent, what conditions must

the coefficients

and the elements s^ of Supp(f) n Some conditions on are easy to establish

satisfy?

(Theorem 10.12).

f^

The main result concerning <s±>

Theorem 10.6, which shows that i .

R

is a monomial.

the first condition implies the second is clear. R[X;S]

it

^f . R[X;S]

R[X;S]

j ,

Supp(f)

is

is a subgroup for each

In this connection, we recall from Section 2 that

is asubgroup of k > 0 .

The

set

S

if and only if s = s + ks

S*

of elements

a group forms a subsemigroup of see this, note that if s + t = s +

S

s = s + k^s

t + k^k2 (s + t) .

statement of Theorem 10.6.

s e S if and

<s>

for some

such that <s>

S*

is

is nonempty; to t = t + k 2t , then

This fact will be used in the

The proof of (10.6) uses two

preliminary results, Theorems 10.2 and 10.4. THEOREM 10.2.

Let

R

be a ring and let

S

be an

additive semigroup. (1) that

Assume that

{s^}^

is a finite subset of

k > 1 is an integer such thatCs^}^ = {ks^}^

for

1 < i < n ,there exists a positive m• such that s. = k 1s . . i l (2) that

Assume that

e = s*L_-|.ri.XSi

R

integer

has prime characteristic

S

and

.

Then

nu < n

p

and

is the canonical form of the idempotent

115

e

of

R[X;S]

.

For each

exist positive integers

i

between

1

and

n , there

s^

= pm is^

m^,k^ < n such that

and

pk i ri = T f Proof. each i

s^

(1):

The equality

= {ks^}^ implies that

is uniquely expressible

between

inductively

1

and

by

s,

n , define

asks^

for some

b^ = i , and

j .

define br + i

= ksu .Since the set (s^)? r r+1 3 1 elements, there exist positive integers h,q with h < q s n + 1 such that s,

= ks, h

b,

=

h

q

= ... = k

b

.

q-h s,

has

n

Then q-h

b _

h +1

Fix

= k

s,

D,

q

h

We conclude that

k q -h s.

= k ^ V ^ s .

= k^s.

bh Moreover,

q - h ^ n

since

{s }n = {ps }!? il il then follow from

and

set

of the multiplicative semigroup of

S

is a

Proof. nu

If

as in Theorem 10.2.

integer

R

k , and for

Therefore each

has prime characteristic

s^ =

k 0

i

To wit, if

, then

are

p

and

in

R .

e = ^ir i^Si Then

is idempotent. Choose km • s^ = p ^-s, for each positive

sufficiently large,

p*cmisi = 0 .

and e £ R .

An alternate elementary proof of (10.2).

R.

p— group, then idempotents of R [ X ;S ] Assume that

.

= {r ^ * . The assertions i 1 l 1 (1) , considering as a sub­

(2)

COROLLARY 10.3.

D ps; 1

(r

of

if

.

1

q < n + 1. 2 p n (2) ,we note that e = e = e= Z^ = 1r^X

To prove Therefore

= s,

bh

(10.3)avoids

m € Z+ is such that p " ^

the use

= 0for each

of

THEOREM 10.4. u

e T , and that (1)

If

A

idempotent in u + q

Assume that U

T

is a unitary ring, that

is the subring of

is a nil ideal of

T

T

generated by

such that

T/A , then there exists

u + A

q € A n U

u . is

such that

is idempotent. (2)

Conversely,

idempotent and Proof. = 0 .

Then

q

(1):

if q e T

is such

is nilpotent, then Choose

k

that u

such that

V k T = (u ) © (1 - u)

.

e U

q

+q

is

.

(u - u 2)k = u^(l - u)k

We first determine the

decomposition of 1 with respect to this direct sum. We k have (1 — u) = 1 — b ,where b e (u) . Moreover, 1 - b k k lc divides 1 - b , where b € (u ) . Thus, the decomposition k k k k of 1 is 1 = b + (1 b ) , where b and 1 - b are k k idempotent. We have u = b + (u — b ) , and to complete \r

the proof, we show that u - b is in A n U . k k k k u — b = ( u - u b ) + (ub - b ) . Moreover,

We have

k k k u - ub = u (1 - [1 - (1 - u) ] } = k k k2 u {1 - [1 - k(l - u) + . . . . + ( - 1) (1 - u) ]} , 2

which is clearly in ub^ -

U n ( u - u ) E U n A ; = (u - 1) [1 -

(u - 1) (1 -

also,

(1-- u)^ ]k =

[1 - ku + . . . +

k k (- u) ] } ,

2

which again is in proof of

(1) .

U n ( u - u ) £ U n A .

This completes the

117

(2): u - u

2

We have

= 2uq + q

0 = (u + q) —

2

- q is nilpotent; say 2 k (u — u ) , it follows that

Expanding

k

is an element of the ideal of sequently, the ideal Let

e

I

of

is idempotent,

(u + q) (1

, and hence

2 V (u - u ) = 0 .

U

U

generated by generated by

be an idempotent generator

sider the decomposition u + q

2

7k k k-i 2 i = - zi= l(i)u C“ u )

u

potent.

(u + q)

T = Te @ T(1

it follows that

u ic

k+1

u

.

Con­

is

for

idem-

I .

- e) of

T .

(u + q)e

and

Con­ Since

- e)

ue

are orthogonal idempotents. Weobserve that k k k+1 is a unit of the ring Te since Tu e = Te = Te .

As

qe

is nilpotent,

it follows that

potent unit of the ring element

u(l — e)

1 — e .

Therefore

element of u(l - e) and

unitary ring

onto

Let

Then

A[X;S]

f

A

h + q

annihilates is an idempotent

u + q = (u + q)e

= e

,

is a nil ideal of the

idempotents of R[X;S] (R/A)[X;S]

are

are in

in

R/A

R

if

.

be the canonical homomorphism of .

h e R[X;S]

If idempotents of such that

is a nil ideal of

R[X;S]

Theorem 10.4 implies that there exists that

The

as we wished to prove.

Assume that

(R/A)[X;S]

R , then we take Since

U ,

if idempotents of

Proof. R[X;S]

R .

.

that is also nilpotent; whence

+ q(l - e) = 0 . Therefore is in

ue + qe = e

u

u(l — e) + q(l — e)

COROLLARY 10.5.

and only

Te , and hence

is nilpotent since

T(1 - e)

q =e - u

ue + qeis an idem-

is idempotent.

Thus

R[X;S]

f(h)

is idempotent.

, part

(1)

q e A[X;S]

h + q e R

are in

and

of such

118 f(h) = f(h + q) 6 R/A . For the converse, assume that idempotents of are in

R/A .

clude that part

If

e

is an idempotent of

e = r + a

(2)

for some

R[X;S]

(R/A)[X;S] , we con­

r e R , a e A[X;S]

of (10.4), it then follows that

.

By-

a , and hence

belongs to the subring of

R[X;S]

r e R , this implies that

e e R , which completes the proof

of

generated by

Since

(2) . THEOREM 10.6.

is a monoid.

Assume that

Let

S*

be

idempotent of Proof. R[X;S]

.

R[X;S] Let

T = n[r^,...,r ] ring

T

<s>

belongs to

e

as

R[X;S*]

and

an element of

FMR— ring.

1

T

and

then using the fact that (T/M )[X;S]

, where

the other hand,

if

potent because

T

as

Part

(2)

T[X;S]

n .

If

onto

r. i M, 1

for each

e e R [ X ;S * ]

R [ X ;S ]

f^

(T/M^)[X;S] for some

.

X

s^ e S* . r^

On

is nil-

We can therefore write

is nilpotent and

generated by

as asserted.

A

i , then

of Theorem 10.4 then shows that

subring of

X , let

is an idempotent of

is a Hilbert ring. h

R . The t*10

of Theorem 10.2 that r^ e M^

q + h , where

T [ X ;S ] , where

has prime characteristic, we con-

A

(2)

.

, and for each

f^ (e)

T/M.

A

elude from part

Then each

Let

be the canonical homomorphism of between

consisting of

nis the prime subring of

family of maximal ideals of

i

S

S

be a nonzero idempotent of

is a Hilbert

Fix

is unitary and that

is a group.

e =

Consider

R

the submonoid of

all elements <s> such that

e

r .

e ,

g e R[X;S*] e

.

belongs to the

g , and consequently,

,

119 COROLLARY 10.7. group.

If

R[X;S]

Assume that

S

is a cancellative semi­

contains a nonzero idempotent

the supporting semigroup of ticular,

S

f

f , then

is a finite group.

In p a r ­

is a monoid.

Proof.

If

G

is the quotient group of

is a unital extension of zero element of

R*[X;G]

T

and if

R , then considering

f

R*

as a non­

, it follows from Theorem 10.6 that

Supp(f) c H , the torsion subgroup of porting semigroup

S

of

f

G .

Thus, the sup­

is a finite group.

by definition, we conclude that

Since

0 e S , and hence

S

T e S is a

monoid. COROLLARY 10.8.

If

R

is a ring and

odic semigroup, then idempotents of Moreover, potent of

if

S

Proof.

Let

R[X;S]

is not a monoid, then

R[X;S]

S

0

is an aperi­ are in

R .

is the only idem-

. T

be a unital extension of

R

and let

S^

be the monoid obtained by adjoining an identity element to 0 S . Since S is also aperiodic, Theorem 10.6 shows that 0 idempotents of T[X;S ] are in T .Therefore idempotents of

R[X;S]

R[X;S]

are in

R[X;S]

n

T .

If

S

n T = R ; in the contrary case,

isa monoid, then R[X;S]

n T = (0)

.

This completes the proof. Under what conditions are idempotents of R ?

in

Corollary 10.8 shows that a sufficient condition for

this to occur is that

S

odic, Theorem 10.9 shows, and

R [ X ;S ]

S

is

aperiodic.

in

the

If

case where

S is not aperi­ R

is unitary

is a monoid, that the question can be reduced to

120 that in which

S

is a group.

THEOREM 10.9. is a monoid.

Assume that

Let

G

R

is unitary and that

S

be the group of invertible elements of

S . (1)

If the only idempotents

R , then each periodic element of (2) of

Conversely,

if

G

Proof. S .

R[X;G]

To prove

The semigroup

Thus in

1 - X* (1)

are in

Assume that

G

contains

a unique idempotent R[X;S] s

(2) .

R[X;S] If

I = S\G

R[X;S]

. Let

e

be an idempotent of

+ e2 ,where

e^ e R[X;G] and

2

R[X;G]

+

Considered R[X;I] R[X;S]

2

(1

2

6^)62 = (1 - e^ ) e2 = [(1 - e^)e2 ] I

is ideal of and

Then

e^ =

e^ and +

is an idempotent of

2

(1 - ej,)e2 = 0 , and

2

+ e2 = 2e2 + e2 . It follows that

2

e2 = e2 =

write

is aperiodic and contains no identity

element, Corollary 10.8 shows that e2 = 2 e ^ 2

S

- e^)e2 = (1— e^)2e^e2

2

Since

; this

as

e2 c R[X;I] .

(2e-^e2 + e2 ) =e , implying that

e 2 = 2e1e2 + e2 . Therefore

R [ X ;I] .

S .

2

2

hence

is invertible.

2

= e1

(1 —

The hypothesis

is a prime ideal of

groups, this sum is direct, and

2

t .

G = S , the assertion is

abelian

e = e^

.

is in

R [ X ;S ] = R[X;G] + R [ X ;I] .

e

if and only

contains each periodic element of

G * S , then

and we have

R

be a periodic element of

t = 0 ; hence

is sufficient to prove If

are in

s

We show that each idempotent of

clear.

G .

is in

R .

is an idempotent in

implies that

S

R[X;S]

(1) , let

<s>

are those of

contains each periodic element

S , then idempotents of

if idempotents of

of R[X;S]

2

e2^

an idempotent °f

R [ X ;I ] , and again

121 we conclude that

— e2 = 0 .

Therefore

e = e-^ e R[X;G]

The result just established in the proof of

(2)

.

of

(10.9) is of sufficient interest to deserve a separate state­ ment . COROLLARY 10.10. a monoid.

Let

I * S

no periodic element. R[X;S\I] If

R

is unitary and

be a prime ideal of

S

S

is

that contains

Then each idempotent of

R[X;S]

is in

. G

is a group, Corollary 10.7 implies that the idem-

potents of

R[X;G]

subgroup of of

Assume that

R[X;H]

G .

are in

H

, where

H

is the torsion

Moreover, a fixed idempotent

belongs to

subgroup of

R[X;H]

R[X;K]

generated by

, where Supp(e)

K .

e = E^r^X^i

is the (finite) In Theorem 10.14 we

determine necessary and sufficient conditions in order that each idempotent of

R[X;G]

should be in

R , but first we

establish two results that are used in the proof of (10.14). The first of these, Theorem 10.12,

is of interest in its own

right. THEOREM 10.12. of

R[X;S] (1)

f

is an idempotent element

. The content ideal

(2) unitary,

Assume that

Assume that then either

unitary, then Proof.

R

c(f)

is idempotent.

is indecomposable.

f = 0

or

c(f) = R .

If If

R

R

is

is not

f = 0 . It is clear that

c(f2 ) £ [c(f)]2 .

Since

f

2

isidempotent, we conclude that c(f)

c(f) = [c(f)]

is idempotent and finitely generated,

.

Because

it is principal

122 and is generated by an idempotent element R

is indecomposable.

and

f = 0

or

and

f = 0 .

If

R

Noetherian ring

R

is

Assume that

R .

Assume that

is unitary, then

c(f) = R ; if

THEOREM 10.13.

e .

e

= 0

or

not unitary, then

1

e = 0

A

is a proper ideal of the oo n 00 n = nn=;[A , B2 = n^B^,. . . .

Define

2

There exists a positive integer R

is indecomposable, then Proof.

If

R*

is Noetherian and

k

B^ = (0)

= Q^n...nQt

where

is

A

R

R* .

Then

R , then

of

mary components

Let (0)

in

is the intersection

R, of

C =consisting of those pri­

such that |C^|

R*

Hence we assume

is unitary.

B-^

If

k.

be a shortest representation of

the subfamily

then

for some

is an ideal of

P^— primary.

P^ ; assume that

6^ = 6^ .

is a unital extension of

without loss of generality that (0)

such that

A

is not relatively prime to

= k^ , where

B^ = (0)and w e 1re finished.

l ^ k ^ ^ t . If If

k-^

k^ = t ,

< t , then

2

either and

B^ = B^

B? ^

where

and the proof is complete, or else

is the C2 >

intersection of a subfamily C .

Since the set

C

2

of

B2 <

B1

C ,

is finite, it follows

2

that

B^ = B^ Since

that

B^

B^ = (0)

for some

R

k .

is Noetherian, the equality

is generated by an idempotent element. if

R

hence

f , then R

implies

Hence

is indecomposable.

Theorem 10.12 shows that if potent

B^ = B,

R

R[X;S]

has a nonzero idem-

contains a nonzero idempotent

has a nonzero unitary subring.

e , and

This observation

shows that the assumption in the statement of Theorem 10.14

123 that the set

C

is nonempty is necessary.

THEOREM 10.14. Let

E

Assume that

be the set of primes

element of order

p

p , and let

unitary subrings of

R .

G

is an abelian group.

such that C

G

contains an

be the family of nonzero

Assume that

C

is nonempty.

The

following conditions are equivalent.

of

(1)

The idempotents of

(2)

For each

p e E

R[X;G]

and each

are those of T e C , p

R .

is a nonunit

T . Proof.

order

p

ment

If

(2)

fails, then there exists

and a nonzero subring

e such that

T

of

R

g e G

of

with identity ele­

pe

is a unit of T .Part (2) of -X p -1 ig Theorem 10.1 then shows that (pe) E . eX is an idemr

potent of

R [X;G ]

(2) f = e

=

(1) :

R .

We consider a nonzero idempotent

€ R[X;G]

and seek to prove that

is an idempotent generator for c(f)

each e

=>

i=0

that is not in

i. and if

Hence if

H

f e R .

, then ef^ =

is the subring of R

T = n[f1,...,fn ] , then T e C

If f^

generated by and T

is

Noetherian. Moreover, the subgroup is finite, and

H

f e T[X;H] .

of

G

generated by

Each Noetherian ring is a

finite direct sum of indecomposable rings. T = T ]_©... @Tr , where each

n ^Si^x

T^

Thus, write

is nonzero and indecom­

posable. With respect to the decomposition T[X;H] = r E^ © T^[X;H] , the idempotent f has a decomposition f = E^f^ , where each f^ e T ^

for each

f^

i , then

is idempotent.

We note that if

f e T £ R .

Also, since each

for

124 T^

is unitary,

|H|

(2) implies that each prime divisior

is a nonunit in each

therefore sufficient (*)

.

To establish

to prove

Assume that

R

To prove H ^©..

(*) ,

is

is a nonzero indecomposable

that each prime divisor of R[X;H]

(1) , it

the following statement (*) .

Noetherian unitary ring and that

the group ring

of

H

|H]

is a finite group such

is a nonunit of

R .

Then

is indecomposable. we express

H

as a direct sum

of cyclic groups of prime-power order, and we use y

induction on where R

p

k .

is

a nonunit of

containing p

of

R[X;H]

potent of

and

H

10.3. m

Thus m

(2)

(j)(f)

<J>

is cyclic of order

Let

M

.

If

4>(f)

f

is a nonzero idem-

is idempotent in

* 0 . The field

R/M

<(>([£) = 1 + M

00 t M ^ = n-^M

indecomposable Assume that

00 t

M2 =

0 .Hence if k =

c(f) = R ,

•

f = 1

for each .

p

by Corollary

m e M[X;H]

is idempotent, me M^fXjH]

m =

(R/M)[X;H] .

hascharacteristic

f = 1 - m , where

€ M2 [X;H ] ,where

,

be a maximal ideal of

of Theorem 10.12 shows that

m e M-jJ X jH] , where

p

be the canonical homomorphism

p— group. Therefore

Whence

clude that

H

, then

H

R .

(R/M)[X;H]

R[X;H]

is a

= 1 — f

and let

onto

Moreover, part and hence

For k = 1 ,

.

Since

t e Z+ .

Similarly,

By Theorem 10.13,. we con­ , and R[X;H]

is

1 .

R [ X ;H]

is indecomposable for

is the direct sum of

r + 1 • of prime-power order, where

If

cyclic groups H ^ , . . . Hr + ^ . c = P > then we consider

R[X;H] as the group ring of H^ + 1

over

K = H1©...©Hr .

R [ X ;K ]

By assumption,

k = r .

R[X;K]

, where

is indecomposable.

125 Since

K

is finite,

gral extension of implies

p

R[X;K]

R .

is Noetherian and is an inte­

Consequently,

is a nonunit of

R[X;K]

yields the desired conclusion that

p

.

a nonunit of

The case

R[X;H]

R

k = 1

then

is indecomposable.

This completes the proof of Theorem 10.14. The analogue of Theorem 10.14 for monoid rings is the following result. THEOREM 10.15.

Assume that

S

is a monoid.

the family of nonzero unitary subrings of that

C

is nonempty.

Let

C

be

R , and assume

The following conditions are equiva­

lent . (1)

The idempotents of

R[X;S] are those

of

R .

(2)

The idempotents of

T[X;S] are those

of

T for

each

T e C . (3)

The set

G

of periodic elements of

group of

S ,and if the prime

element of

G

Proof.

, then

p

The equivalence of

(2)

and

is a sub­ of an

for each (3)

T

e C .

follows from

It is clear that

Cl)

im­

(2) , and the reverse implication follows from Theorem

10.12:

if

f c T[X;S]

f

is a nonzero idempotent in

for some

COROLLARY 10.16. R

is the order

is a nonunit of T

Theorems 10.6, 10.9 and 10.14. plies

p

S

R[X;S]

, then

T e C . Assume that

S

is a monoid and that

is a unitary ring of nonzero characteristic

n .

The

following conditions are equivalent. (1)

The idempotents of

C2)

Either

(i)

S

R[X;S]

are those of

is aperiodic or (ii) n = p

R . lr

is a

126 prime power and the set of periodic elements of

S

is a

p-group. Proof.

That

(2)

implies

(1)

follows from Theorems

10.6 and 10.15, for the characteristic of each subring of is a divisor of the characteristic of assume

(1)

ments of

S

Theorem 10.6

of

For the converse,

and assume that the

set G of periodic ele­

is nonzero —

is, S is not aperiodic.

shows that

that G

is agroup.

that is theorder of anelement of plies that

R .

p

G

Let

p

be a prime

. Theorem 10.15 im­

is a nonunit of each nonzero unitary subring

R , and this implies that

n

is a power of

p , for

has a nonzero unitary subring of characteristic prime—power divisor k , (10.15) then

d >

R[X;S]

1

of

implies that G

For a unitary ring that

R

R

n

.

is a

d

for each

Since

n =pforsome

p— group.

and a monoid

S , it is clear

is indecomposable if and only if

decomposable and the idempotents of

R

R[X;S]

R

is in­

are those of

R .

Since the identity element of each nonzero unitary subring of an indecomposable

unitary ring is the same as the identity

element of the ring itself, the next result follows from Theorem 10.15. COROLLARY 10.17. S , the ring

R[X;S]

indecomposable, subgroup of ment of

S

R

and a monoid

is indecomposable if and only if

the set of

S , is a

For a unitary ring

periodic elements of

and the order

of

nonunit of

.

R

S

R

is

is a

each nonzero periodic ele­

The characteristic of an idecomposable ring is either

0

127 or a prime power.

In the case of a ring of prime-power char­

acteristic, Corollary 10.17 translates to the following statement. COROLLARY 10.18.

Assume that

is an indecomposable v unitary ring of prime-power characteristic p . Let S be a monoid, and denote by S .

Then

R[X;S]

p— subgroup of

S*

R

the set of periodic elements of

is indecomposable if and only if

S*

is a

S .

We conclude the section with a result for group rings that has the familiar ring of (10.14) — (10.18). THEOREM 10.19.

Assume that

unitary ring and that primes Gp

p

G

potent of

R

If

and let

is in

R[X;H]

f e R[X;G]

Gi = Hi @ H 2 , where for each

group ring of f e R[X;H^] R

H^

H2

over

B

be the set of

H = ^peB^p * where G .

Then each idem-

.

is idempotent, then

for some finitely generated subgroup

(H2)p = (0)

Let

p— primary component of

R[X;G]

Proof.

is an indecomposable

is a group.

that are units of

denotes the

R

G-^

of

is a subgroup of

p e B . RfXjH-jJ

f e R[X;G^]

G . H

Considering

Moreover,

and where R [ X ;G ^ ]

, Theorem 10.14 implies that

if each prime integer

q

that is a nonunit of

is also a nonunit of each nonzero unitary subring

R[X;H^] that

qe

.

Let

e

be the identity element of

is a unit of

contents of

e, qe , and

T — t

of Theorem 10.12 shows that

say

as the

T

e = qe • t .

as elements of c(e) = R .

qc(e*t) 5 qR t contrary to the fact that

R[G]

But q

T

of

and assume Computing , part

(2)

c(qe*t) = is a nonunit of

128 R .

This completes the proof. Section 10 Remarks The literature is rich in material concerning idempotents

of group rings ring and

G

R [G ] , where

is nonabelian.

R

is a commutative unitary-

Two general references for some

of this material are Chapter 1 of [121] and Chapters 2 and 4 of [125]. whether

It is unknown in the nonabelian case, for example, R[G]

has only trivial idempotents if

composable and

p

R

is inde­

is a nonunit of

R

for each prime

p

that is the order of an element of

G

(cf. Conjecture 2.18

of [125, p. 23]); Formanek in [43] has proved some results related to this question. There is an analogue of (10.15) for semigroups are not monoids.

S

that

It involves use of Theorem 10.6 and intro­

duction of the family of subsemigroups of

S

that are

monoids. In general, presentation of the material in this section has followed that of Section 2 of [61]. The description of oo ft nn _^A used in the proof of Theorem 10.13 can be found in Section 7, Chapter IV of [136]. The condition in (10.14) and (10.15) that

p

is a n o n ­

unit of each nonzero unitary subring of

R

the condition that the ideal

contains no nonzero

idempotent.

pR

of

R

is equivalent to

129 §11.

Units

All rings considered in this section are assumed to be unitary, and all semigroups are assumed to be monoids. area of units of monoid rings is vast — the topic of a monograph in itself —

The

large enough to be

and hence some restric­

tions on the scope of treatment here are necessary.

The main

restriction we impose is in considering primarily torsion—free monoids.

In this case, we are interested in characterizing n s, the units u = E^r^X of R[X;S] in terms of the coeffi­ cients

r^

and the elements

a description of

s^

of

Supp(u)

the Jacobson radical of

case.Another problem of significant

.

We also seek

R[X;S]

in this

interest is that

giving conditions under which each unit of

R[X;S]

of

is

trivial, where by a trivial u n it , we mean one of the form rXs

where, necessarily,

vertible in

S .

r

is a unit of

R

and

s

To begin, we consider the case where

is in­ S

is

both torsion— free and cancellative. THEOREM 11.1. with identity and

Assume that S

D

is an integral domain

is a nonzero torsion— free cancellative

monoid. (1) then

f

If and

f,g e D[X;S]\(0) g

fg is a monomial,

are monomials.

(2)

D [ X ;S ]

admits only trivial units.

(3)

D[X;S]

is semisimple.

Proof.

(1):

Let

<

with the monoid operation. deg(fg) = deg(f) + deg(g) and

and if

ord(g) = deg(g) —

be a total order on Since

S

compatible

ord(f) + ord(g) = ord(fg) =

, it follows that

that is, f and

ord(f) = deg(f)

g are monomials.

130 Statement

(2)

follows from (1) . To prove (3) , n sa take a nonzero element h = E-^r^X in D[X;S] . Since S is infinite and

cancellative, there exists s e S

0 ^ Supp(XSh) D[X;S]

. Consequently,

, and

1 + X Sh

such that

is not a unit of

his not in the Jacobson radical

of

D[X;S]

.

The statement of Theorem 11.3 uses one equivalence which is purely— ring theoretic,

and which we choose to state as a

separate result. THEOREM 11.2. unitary ring

n

Let

R.

^a i'i=i

be a finite subset of the

The following conditions are equivalent.

(1)

R = (a^,...,an )

and

a ^aj

is nilpotent for

(2)

There exists a positive integer

i * j k

such that

ak )®...©(ak) . 1 n (3) that

There exists a finite subset

E^a.b. = 1 I i i

Proof.

and

a.b. i

(1) = >

of

is nilpotent

j

for

r

(2):Choose

k

R

such

i * j . J

large enough so that

Y

(aiaj) R =

= 0

forall

k k (a1 ,...,an ) . (a^) n i

Therefore (2)

i * j .

Moreover, for E

(a^) =(

j n

(3): For

a

]

R = (a^)@...©(a^) =>

Since

i

R

= (a^,...,an ) ,

then

1 £ i £ n ,

j

.(a^) *

i

= (0)

j

.

.

1 £ i £ n , let

a^ = E ? =1c —

be the

decomposition of a^ with respect to the decomposition n k k n k R = £j = 1@(aj) of R . Then a i = Zj.jCjj implies that k k v c^i = a^ and c^j = 0 for each j * i . Therefore c^j ring

is nilpotent for (a-j^) #

Let

b^

i * j

and

c^

is a unit of the

be the inverse of

c^

in

(a^^)

131 for each

i .

Then

Moreover, (3)

b^a^ = b ^ c ^

= b^c^ — > (1):

is nilpotent for j * i .

> so

= ^b^c^ = 1 .

The equality

^a^b^ = 1

implies that

R = (a^,...,an ) , and multiplication of the equation by for

u * v

biau

or

yields auav = ^i = ia ibiauav . b^ay

is nilpotent.

For each

Therefore

auav

i

au av

either

is also nil-

potent . THEOREM 11.3. N

and with set

Let

R

be a unitary ring with nilradical

^x^AeA

ProP er prime ideals.

a torsion— free cancellative monoid, invertible elements of

f e R[X;S]

for each

f

A , let

be

onto

de­ (R/P^)[X;S].

in the form

g = ^i=igiXSi

of

i

g ,

Supp(g) n Supp(h) = <J> .

S

be the group of

R [X ;S ]

, and write

where the canonical form g^ k N

G

S , and for each

note the canonical homomorphism of Assume that

let

Let

g + h ,

is such that

h =

* 6 N[X;S]

The following conditions are equiva­

lent . (1)

f

is

a unit of R[X;S]

(2)

g

is

a unit of

(3)

<J> (g) is a unit of (R/P )[X;S] for each AeA A A c(g) = R , gigj e N for i * j , and Supp(g) c G .

(4)

R[X;S]

. .

(5) There exists a positive integer k k R = (g^©. • •®(gn ) and Supp(g) c G . (6) R

Supp(g) c G

such that

and there exists

1 = E^b^g^

» where

b^g^

k

such that

asubset is

of nilpotent for

i * j • Proof.

The equivalence of

and the equivalence of

(4),

(5),

(1) and and

(2) (6)

is well known, is the content

,

and

132 of Theorem 11.2. (3)

=>

The implication

(4):

Fix

X e A .

(2) = >

(3)

is clear.

Theorem 11.1 shows that

(g) = (g.)X is a trivial unit of (R/Px)[X;S] . It X IX l follows that there exists a unique i between 1 and n such that

»an(*^or t h i s

gi ^ p x

c(g) = (g1 ,*.*,gn ) To see that each exists a prime since

4>a (g) (6)

s^ Pa

eG

= > (2):

, s^ e G .

andis nilpotent , note that

of R

such

is a unit of

w = ^ 1 = 1 b^X Si . i * j

=R ,

i

Therefore for

i * j

g^ | N implies

that

gfc ^ pa ‘

there

T^en

s^ e G

(R/Pa)[X;S]'.

Under the hypothesis of The hypothesis that

(6) , we

b^gj

let

is nilpotent for

implies that gw = (EgiX551) (IbjX-51) = (Ejgibi) + v = 1 + v ,

where

v e N [ X ;S ] .

Therefore

gw

is a unit, and

g

is

also a unit. The statement of Theorem 11.3 can be simplified in certain cases where it is unnecessary to pass from decomposition

f

to the

g + h .

COROLLARY 11.4. Assume that

R

is a unitary ring and

that

S is a torsion— free cancellative monoid. Let n s• f = E^a^X 1 be the canonical form of f eR[X;S], and let

G

be the group of invertible elements of (1)

if

if

IfR

and only

R = (a )@...©(a ) and Supp(f) c G . I n ~ (2) IfS is a group, then f is a unit if

and only

(3)

If

and R

reduced, then

a^aj

f

is

S . a unit if

c(f) = R

is

.

is nilpotent for

is indecomposable, then

i * j . f

is a unit if and

133 only if there exists

i

is a trivial unit of

R [X ;S ]

COROLLARY 11.5. nilradical

N

between

cellative monoid.

S

radical of

R

is a unitary ring with

is a nonzero torsion— free can­

The Jacobson radical of

N [X;S ] , the nilradical of Proof.

and

and

Assume that

and that

s. n such that a^X 1 s• f - a^X 1 is nilpotent.

1

R[X;S]

.

By passage to

N[X;S]

is the nil-

(R/N)[X;S]

, we can

prove Corollary 11.5 by showing that if

R

is reduced.

and choose Xsf * 0

s e S

since

Xs

0 ^ Supp(XSf)

(1)

R[X;S] .

only trivial or

(2)

Let

R[X;S]

.

It

1

Consequently, f^ J(R[X;S])

R, S,

and

+ X Sf and

(2)

G = (0} .

or

(2)

described

R

in

R [X ;S ]

has

is reduced and either

is indecomposable.

It is immediate from part has only trivial is satisfied.

is a nonzero nilpotent of unit of

be as

is satisfied:

R

R[X;S]

G

The monoid ring

units if and only if

(1)

Proof.

(1)

f e R[X;S]

of Corollary 11.4 that

the statement of Theorem 11.3.

that

semisimple

is semisimple, as we wished to show.

THEOREM 11.6.

(1)

is

; we note that

is a regular element of

then follows from part

R[X;S]

R [ X ;S ]

Thus, take a nonzero element such that

is not a unit of

is

.

Theorem 9.9 implies that R[X;S]

R[X;S]

R[X;S]

other hand, if

If

(1)

of Corollary 11.4

R

is reduced and

units if R

R , then

is not

1 + rXs

for each nonzero element R

reduced and if

is decomposable and

s

r

is a nontrivial of

S .

On the

G x {0} , then take

134 a decomposition potents of Then

R

1 = e^ + e 2

and consider a nonzero element

e^ + e 2X °

verse

int0 nonzero orthogonal idem­

a nontrivial unit of

e^ + e 2X ^ .

g

of

R[X;S]

G .

with in­

This completes the proof of Theorem 11.6.

In seeking to extend Theorem 11.4 and its consequences, we have a choice of weakening the hypothesis that torsion— free or the assumption that our purposes, hypothesis;

S

S

is

is cancellative.

For

it is more reasonable to weaken the cancellative

as soon as one drops the assumption that a can­

cellative monoid is tors ion— free, then the quotient group of S

admits a nontrivial finite subgroup

of determining the units of

R [ X ;H ]

H , and the problem

arises.

This is an in ­

teresting problem that is not at all intractable, but it's also a very large problem area that has limited applicability to topics subsequently covered in this text. the assumption that

S

is tors ion— free.

Hence we retain

To give an indica­

tion of the generalization of Theorem 11.4 that we seek, we reinterpret that result as follows. is a prime ideal of

S

Each

f € R[X;S]

f^ e R[X;G]

where

G— component of

f

and and

as

f'2 the

I— component

the I— and I

(0) .

^ of

S .)

If

unit of

if and only if

torsion— free and cancellative,

f 2 e N [ X ;I ] , where

N

the f .

f^

(More of

f

G = S , then we

In this terminology,

S

and

f^ + f2 ,

(S\I)— components

of

states that for R [X ;S ]

I

is an ideal of

has a decomposition

for any proper prime ideal R [X ;I ]

R [ X ;I ]

f2 e R [ X ;I ] ; we call

generally, we speak of

interpret

I = S\G , then

and the monoid ring decomposes as

R [ X ;S ] = R [ X ;G] + R[X;I] , where R[X;S] .

If

is a unit of

is the nilradical of

(11.4) f

is a

R[X;G] R .

In

135 Theorem 11.13, we extend this result to the case where torsion— free and aperiodic.

S

Note that a torsion— free

is

can­

cellative monoid is aperiodic, for its quotient group is torsion— free.

The converse fails.

S = Z q © Z q @ Z q by

and if

(a,b,c) ~ (d,e,f)

then

S/~

~

if

For example, if

is the congruence on a = d > 0

and

S

defined

b + c = e + f ,

is torison— free and aperiodic, but not cancella­

tive; to within isomorphism,

S/~

can be thought of as the

iilc x yJ z

multiplicative monoid

of terms

R[X,Y,Z]/(XY - XZ)

The next result gives an equivalent

.

in

description of a torsion— free aperiodic monoid. THEOREM 11.7. if and only

if

Proof.

The torsion— free monoid

0

S

is the only idempotent of

Clearly

0

For the converse, assume that

idempotent of

S

The semigroup

<s>

that

and let

s

Since

of 0

S

if

S

S

is the only

be a periodic element of

contains an idempotent

by hypothesis.

S.

is the only idempotent

is aperiodic.

0 = kO

is aperiodic

ks , and

S

.

ks =

is torsion— free, it follows

s = 0 . If

S

contains a nonzero idempotent

11.4 need not extend to the monoid ring example,

1

- 2

is not nilpotent in

.

For

2

and

Z .

Theorems 11.9 through 11.12 is a special case of

Theorem 11.13.

We assume throughout these results that

is a unitary ring, monoid,

R[X;S]

- 2XS e Z[X;S] is a unit of order

s { G , but Each of

s , then Theorem

and that

that G

S

R

is a torsion— free aperiodic

is the set of invertible elements of

S .

For the sake of brevity, we do not repeat these hypotheses in

136 the statements of (11. 9)-( 11.12) .

The proof of (11.9) uses

an auxiliary result that has already been proved in

(2)

of

Theorem 1.1; we repeat the statement of the result in the form in which it is used here. THEOREM 11.8. monoid Then

S I

Assume that

and that

I

is contained in an ideal

p * 0

istic

of

disjoint from S

T .

maximal with

T , and each such

Assume that

and

assume that

proper prime ideal of where

f e R[X;S*] Proof.

exists

S

.

, then

t e S*

J

is a prime

S*

I

t + kt = kt

I

is empty. 1=

{s e

, then

If not,

S |t + s =

is a proper ideal of



of

A

S , it S .

Therefore

[s.], [t] £ T

ns + c = nt + c

n(s + c) = n(t Thus

S

such that the We prove that are such that

for some

+ c) and

[s] = [d]

S

is empty.

is cancellative.

Thus, if

s}

S ,and

k e Z+ , contrary to the fact that

T = S/~

,

A = {s e S | there

be the smallest congruence on

is torsion— free. [x]

such that

We conclude that

is torsion— free.

Therefore

is a unit ofR[X;S]

meets the semigroup

factor semigroup

n[s] = n[t]

1+f

unique proper prime ideal of

for some

is aperiodic. ~

S* = S - {0} is the only

s = s + t}

t e S*

is the

follows that

Let

is a field of character­

f = 0 .

such that

is also nonempty.The set since

If

R

We first claim that the set

then there exists

Let

J

S

S . THEOREM 11.9.

T

is a subsemigroup of the

is an ideal of

respect to failure to meet in

T

c

s + c = t + c and

be an invertible element of

T

£ S . since

is tors ion— free.

T , say

S

137 [0]

= [x] + [y] = [x + y]

.

is empty, we conclude that

Then

x + y

x + y = 0 .

~

0 , and since

Because

S*

only

proper prime ideal of

S , it follows that 0

only

invertible element of

S .

[0]

is the only invertible element of

Therefore

is the is the

x =y =

T .

A

0 , and

Summarizing,

T

is a torsion— free cancellative monoid with no invertible ele­ ment other than

0 .

the only units of

It then follows from Theorem 11.6 that

R[X;T]

are those of

the canonical homomorphism of <J>(1 + f) = 1 + (f) e R • is empty.

Therefore

s^,sj e Supp(f)

R[X;S]

But

R[X;T]

Let

s^ + t .j = Sj + t^. . Let

over

all

<J>

is

, then

since

f = E^r^Xsi .

s^~ s^. , then we choose

that

s^ , s. e

onto

Thus, if

[0] ^ Supp(<J>(f))

<J>CfD = 0 •

and if

R .

A

If

t ^ e T such

t= £t^. > where the sum is taken

Supp(f)such that

s^ ~ s^. . It

then

follows that consider

s. ~ s. if and only if s- + t = s. + t . We 1 J 7 1 J X tf = Z^r^Xsi+t = Z^a^X^ , where bj,...,bm are

distinct and where

b. / b. 1

for

i * j . Then

3

0 = <|)(X1")4> (f) = (j)(Xtf) = z^a^X^j-’ 1 * j , it follows that In particular,

f = 0

Assume that of

1 + f

and since

b^ /- b^

for

a 1 = ...= am = o . Therefore Xtf = 0 . if

t = 0 .

t * 0 . It is easy

is of the form 1 + g

for

to see that the inverse some

g e R fX;S * ] .

The contrapositve of Theorem 11.8 implies that each ideal of S <s> p^s -so

meets

<s>

for each € t + S 0 = f + g

By choice of Tjk rjk f g = 0 .

for

s e S*.

s e Supp(g) . for each + fg= k , Xt

In particular, Choose k

t+ S

meets

large enough so that

s e Supp(g) . Then 1 = (1 + f) (1+ g) Tjk r»k nk nk nk ( f + g + fg)P = fp + gP + fP gp . is afactor of

Consequently,

fF

gPk , and hence nk = “ g and

,

138 "y k

k k = _ fP gP = o .

f P

that the ring

R[X;S]

THEOREM 11.10. istic

p * 0

R[X;S]

, where

Proof.

Hence

and

f = 0 , for Corollary 9.3 shows

is reduced. Assume that

that

R

G = {0 } .

is a field of character­

If

1 + f

f € R[X;S - G] , then

is a unit of

f = 0

.

Assume that Theorem 11.10 is false.

(1 + f) 1 = 1 +

g , where g e

that

+ ISupp Cg)I

I Supp(f)l

R[X;S - G]

.

S

We may assume

is minimal among all counter­

examples to the statement of the theorem. the submonoid of

We write

generated by

Replacing

U = Supp(f)

S

by

u Supp(g)

, we

also assume without loss of generality that the subsemigroup of

S

generated by

it follows that

U is

S*

S* = S \ {0}.

Since

is a prime ideal ofS .

S* = S\G

From

Theorem

11.9, we conclude that there exists a prime ideal properly contained in As

write

1 + f

and

1 + g =

f2 (1 + g-j^) = 0 . R[X;S\P] f2

I Supp(f)| g!

or

U £ P

1 + f^

Thus

is the

1 + f.^

S

1 + g^ .

and

1

.

We

(Subcom­ 1 + f .

is the =

+ f)g2 +

is a unit of the Since

monoid

U & S*\P ,

ISupp f-jJ + ISupp g-jJ U t

P , either

1 + g^

provide a

Moreover, as

1 + f^

1 + g^

and (1

is nonzero, and .

U ± (S*\P)

Itfollows that

fx)(l + gl)

with inverse g2

and

is the P— component of

Therefore

+ lSupp(g)|

is nonzero.

f2

1 +g .

(1 + f)(l + g) = (1 +

of

is generated as a semi­

1 + g^ + g 2 , where

(S\P)— component of

either

S*

l + f = l + f 1 + f2 , where

Similarly,

ring

and

U , it is clear that

ponent of

P

S* .

S* = (S*\P) u P

group by

,

< f^

or

coun­

terexample to Theorem 11.10 that contradicts the *assumption

139 that

ISupp(f)|

+ lSupp(g)|

is minimal.

This completes the

proof of Theorem 11.10. THEOREM 11.11.

If R

is

p *' 0 , then each unit of Proof. and write are the

Let

R[X;S]

be a unit of

is in R[X;S]

R[X;G]

G— components of

1 +

R[X;G]

1 + v xu 2

and

and

u iv 2

R[X;S]

are

u^

and

v , respectively.

with inverse

is a unit of v xu 2

u

.

with inverse

u = u^ + u 2 , v = v^ + v 2 , where

is a unit of

As

u

a field of characteristic

v^ , and hence

with inverse

R[X;S\G]

v v^

Then

u^

v^u =

u^v = 1 + u^v2 .

, it follows that

is, in fact, a unit of the monoid ring

R[X;(S\G) u {0}] . 0 = v^u2 ,

Applying

and hence

u2 = 0

Theorem 11.10,we

conclude that

.

u^ e R[X;G]

Therefore

u =

as was to be proved. THEOREM 11.12., of

R [X ;S ]

is in

Proof. verse let

Let

R[X;G]

is

a reduced ring, then each unit

.

f = E^f^X5 *

g = E^g^Xt i . Let

IT

T = n[ {f± }, {g± >] . Then

Let

be a unit of

R[X;S]

A € A , let <J>.

T

is a Hilbert

onto CT/MX )[X;S].

for each

A .

h e

R

and

FMR-ring. T

and for

be the canonical homomorphism of

A

TTX;S]

with in ­

be the prime subring of

t*ie family °f maximal ideals of

each

then

If R

Thus, if

ByTheoremll.il, h

is the

(M^[X;S\G]) = N[X;S\G]

nilradical

of. T .

Because R

f e R[X;G]

as asserted.

*x(f)e(T/Mx)[X;G]

(S\G)— component of , where

is reduced,

f

N is the N = (0)

and

Theorem 11.12 is the last preliminary result needed for

,

140 the proof of Theorem 11.13. hypotheses on

R

and

S

In stating (11.13), we repeat

that have sometimes been assumed

implicitly in the statements of (11.9)— (11.12). THEOREM 11.13. that

S

Assume that

is a unitary ring and

is a torsion— free aperiodic monoid.

nilradical of

R

ments of

An element

if the

R

S .

and let

G— component of

S\G— component of Proof.

f

G

R[X;G]

The only part of the statement of Theorem 11.13

nent of a unit

f

is in

under the canonical map

N[X;S\G] <J>

onto

(R/N)[X;G]

by Theorem 11.12.

of

Then

nent of

g

that

and

the

f

and the

N[X;S\G].

that requires proof is the assertion that the

<J>(f) •

be the

is a unit if and only

is a unit of

is in

N

be the group of invertible ele­

f e R[X;S]

f

Let

N[X;S\G]

.

have the same

(S\G)— component of

f

The image

(R/N) [X ;S ]

Since

(f)

f

, so the

be a preimage (S\G)— compo­

f = h + g , it follows

(S\G)— component.

is in

of

is in

h e R[X;G]

Let

f - h = g e N[X;S]

is in g

.

(S\G)— compo­

N[X;S\G]

Therefore

, as asserted.

Corollary 11.5 and Theorem 11.6 extend easily to the case where

S

is torsion— free and aperiodic.

We state these

extensions without proof. THEOREM 11.14.

If

R

is unitary and

S * (0)

is a

torsion— free aperiodic monoid, then the Jacobson radical of R[X;S]

is

N[X;S]

, where

N

is the nilradical of

Thus, the nilradical and the Jacobson radical of

R .

R [X ;S ]

coincide. THEOREM 11.15.

Let the hypothesis on

R

and

S

be as

141 in Theorem 11.14.

The monoid ring

units if and only if

R

R [X ;S ]

has only trivial

is reduced and either

(1)

or

(2)

is satisfied. (1)

R

is reduced.

(2)

G = {0}

.

We return briefly to the considerations in Theorem 11.13 to develop a notion that will be used in Section 22 in exam­ ining the

R— automorphisms of

R [X ;S ] , and in Section 24 in

considering the relationship between rings the monoid rings

R ^ [X ;S ]

are isomorphic.

Let

of Theorem 11.13.

and

R, S,

If

R 2 [X;S]

and

1 = E^e^

into orthogonal idempotents

e^

G

lowing: ^lU i

if

E

u

splits

of

S ,

and if

u

1

is a unit of

E =

splits s^,...,sn e G w

u

if

(not

such that

The value of this concept is the fol­ u , then in the decomposition

with respect

© Re^[X;S]

if

be as in the statement

v € R , elements

+ w .

R2

, for a fixed

necessarily distinct), and a nilpotent element u = v(Z^e^Xsi)

and

is a decomposition of

R [X;S ] , then we say that the set there exists a unit

R^

R[X;S]

to the decomposition , each u^

R[X;S]

u

=

=

has the form

(trivial unit of Re^[X;S]) + (nilpotent element of Re^[X;S])

.

A unit of this form is frequently easier to work with than a general unit.

In Theorems 11.16 and 11.17 and in Corollary

11.18, we continue to use the hypothesis of Theorem 11.13 on R, S, and

G ; thus

aperiodic monoid, and of

S .

R

is unitary, G

S

is a torsion— free

is the group of invertible elements

142

THEOREM 11.16. units of

R[X;S]

identity of

R

Assume that

.

u =

that

G , h

=

splits each

+ h

in

u^

k e Z+ .

i , where

For

k = 1, take

is nilpotent.

into orthogonal idempotents

e^

s^

if

a is in

By Theorem

1 = ei+ -*-+en

such that

Moreover,

of the

.

k .

there exists a decomposition

each

E

R[X;S] , where each

is nilpotent, and no

11.3,

*s a finite set °f

There exists a decomposition

Proof. We use induction on unit

^u i^i=i

°f

*

Re^ = Rg^

for

g^ =

is the

decomposition of

g^

R =

R , then the proof of Theorem 11.2 shows

@ Re^

that

g^

of

is a unit

i * j .

Therefore

potent.

Let

v =

sequently, For

{e^fj

of

Re^

, while g ^

u =

k = 2 , it and

v

is easy to show that if E splits u ^ > then

F = {f^

splits

both

It follows by the same reasoning that if and

wise products

F

splits

ef

, for

is nil-

R

+ h* = u

for

and .

Con­

u .

| 1 < i
u^,...,^

h*

is a unit of

^lveiX Si+ h*= Z^giiXSi

E = {e^splits

u^

is nilpotent

+ h* ,where

Ejg. . . Then

v C ^ eiX si) + h* =

splits

with respect to the decomposition

uT+i

ee E

E

= {e^

B u^

= and

.

splits

> then the set of all pair­ and

f e F , splits

each of

U l>*-- ,ur+l * THEOREM 11.17. the identity of split by R [X ;S ] .

E

R .

Let

E =

be a decomposition of

The set of units

u e R[X;S]

that are

is a subgroup of the multiplicative group of

143 u = v(Ee^Xsi) + w E , then hence

uu*

sentence.

and

u* = v*(Ze^Xti) + w*

are split by

uu* = vv*(Ee^Xsi ft;*-) + wu* + w*v(Ze^Xsi) , and is split by Let

N

E .

Let

u

be as in the preceding

be the nilradical of

the canonical homomorphism of

R[X;S]

R onto

and let

<J> be

(R/N)[X;S]

.

Then *( U_1) = [4> (u)]"1 = [♦(v)J^Ce.)XsI]'1 =
.

u*1 - v " * (Ee^X"3! ) e N[X;S]

COROLLARY 11.18.

If

H

and

E

u"1 .

is a finitely generated sub­

group of the multiplicative group of units of there exists a decomposition of of

splits

1

R[X;S]

, then

that splits each element

H . Section 11 Remarks Most of the material in this section on units of

R[X;S]

, in the case where

comes from [55].

Units of

S

is torsion-free and aperiodic,

R[X;S]

of finite order are also

determined in [55]. Since our treatment of units is modest in terms of what is known, it seems appropriate that we give a few references to additional material in the literature. Passman [121] and Sehgal

[125] are general references on

units of (not necessarily commutative) of Higman

[76], Cohn and Livingstone

[16], and May

Again the books of

group rings.

Papers

[34], Ayoub and Ayoub

[102] deal with the group of units of commuta­

tive group rings, with particular emphasis in the first two

144 papers on determining conditions under which a group ring R[G]

has only trivial units.

an abelian torsion group Z[G] G

G

Higman's classical result for is that the integral group ring

has only trivial units if and only if the exponent of

divides either

4

or

6 .

Ayoub and Ayoub determine the

structure of the group of units of

Z [G ]

for a finite group

G , as well as the structure of the group ring the field

Q

0 [G]

over

of rational numbers.

Our restricted treatment of units of

R[X;S]

in this

section has some consequences in Section 25, where the ques­ tion of whether isomorphism of implies isomorphism of

S^

R[X;S^]

and

S^

and

R[X;S2 ]

is considered.

Partly

because some needed results concerning units are not avail­ able in the text, Section 25 is merely a survey of some of the known results on the isomorphism problem; no proofs are presented in Section 25. For a periodic monoid

S

and a unitary ring

Gilmer and Teply in [60] determine

(1)

R [S]

is in

for which each unit of

R[S]

is the group of invertible elements of monoid rings

R[S]

R ,

those monoid rings R[G]

S , and

with only trivial units.

, where (2)

the

G

CHAPTER III RING-THEORETIC PROPERTIES OF MONOID DOMAINS

For a number of ring— theoretic properties natural to investigate conditions on the semigroup ring

R[X;S]

Theorem 7.7 shows that if then and

R[X;S] S

R

and

has property R

E , it is S

E .

is unitary and

is Noetherian if and only if

under which For example,

S R

is a monoid, is Noetherian

is finitely generated; Theorem 9.17 gives equivalent

conditions under which an arbitrary semigroup ring is reduced. In Chapters III and IV we consider several other ring— theore­ tic properties

E .

The conditions considered in Chapter III

are of interest primarily for integral domains — integral closure,

for example,

the factorial property, and the conditions

that a domain should be a Priifer domain or a Krull domain.

In

Chapter IV we consider properties that are also of interest in the case of rings with zero divisors. In contrast with Chapter II, restriction to the case of unitary rings of the text.

R

and monoids

In particular,

S

is more common in the rest

in Chapter III

R

assumed to be a unitary integral domain and

S

cancellative monoid,

so that the monoid ring

unitary integral domain. Chapter II notation shorten it to

R[S]

is usually a torsion— free R[X;S]

is a

There is also an abbreviation of the

R [ X ;S ]

in the remainder of the text; we

, for the primary stress will shift from

elements and their canonical representations to concepts more 145

146 global and ring— theoretic in nature. tension ring of the ring

R

and if

(not necessarily closed under R[E]

+

If E

T

is a unitary e x ­

is a subset of

T

or •) , then the notation

is sometimes used in commutative algebra to denote the

subring of

T

generated by

R u E .

between this notation and the notation ring of

S

over

R u { X s |s e S}

.

R , for

R[X;S]

There is a consistency R [S ]

for the monoid

is generated as a ring by

147 §12.

Integral Dependence for Monoid Rings

In the remaining sections of this chapter, we determine equivalent conditions in order that a monoid domain should be factorial, a Priifer domain, or a Krull domain.

A common

feature of the domains of each of these classes is that they are integrally closed.

Hence, this section is devoted to de­

termining the integral closure or complete integral closure of one monoid ring in another, and to determining conditions under which a monoid domain is integrally closed or completely integrally closed.

The nature of the results is such that we

need not restrict to a consideration of integral domains only, although that condition is imposed throughout most of the other sections of this chapter.

We do assume, however, that

all rings and semigroups considered in the section have iden­ tity elements.

And to repeat, the customary notation

of Chapter II is abbreviated to write elements of

R[S]

R[S]

in the form

R[X;S]

here; we continue to Si= iri^S*' *

The first two results of the section are related to questions of integral dependence, but they have much wider application than that.

In fact, various parts of Theorem

12.2 are cited throughout Chapters III—V.

These results are

used primarily to transfer various properties from one monoid ring to a subring or to an extension ring. ring of the ring of

T

T , we say that

R

if there exists a homomorphism

identity mapping on the ring show that

R

B

of

T

R

is a sub­

is an algebraic retract T -- ► R

that is the

It is straightforward to

is an algebraic retract of

there exists an ideal R n B = (0) .

R .

If

such that

T

if and only if T ■ R + B

and

148 THEOREM 12.1.

Assume that

R

is a unitary ring and

S

is a monoid. (1)

R

is an algebraic retract of

(2)

R[S\P]

is an algebraic retract of

proper prime ideal (3)

P of

Assume that

and leta e A cosets of

R [S ] . R[S]

for each

S .

His a subgroup of

S

containing

be a complete set of representatives for the H

in

S

.

Asa module over

R[H]

R[H]XSa

then

In any case, R[H]is a direct

{XSot)aeA

summand of

is free.

R[S]

Proof.

The augmentation mapping on R[S]

tion

of

R[S] = R[S\P]

+ R [P]

an algebraic retract of To prove

(3) ,

ZaR[H]XSa .

sa • 1 = 0 S

To show that f^

for each R[H]

defines

R

shows that

R[S\P]

is

. s = sa + h

for

e R[H]XSa

and

X s = XhX Sa

R[S] =

the sum is direct,assume that e R[H]

i

if

i . S

sentative of the coset R[H]

R[S]

for each

i .

J ) = for

By choice of

i * j .

Hence

sa for each

is cancellative,

over

is cancellative,

Moreover, the decomposi­

R[S]

S qj . s oi. , Suppff.X -1) n Supp(f.X l j

f^ = 0

so

.

take s e S .Then

E^f^X a i = 0 , where

If

R[S]

a e A, h e H .Hence

{s } a

S

is the

.

as an algebraic retract of

f^X

; if

, R[S]

direct sum of the modules

some

0

and the sum EaR[H]X then

Thus

XSa is regular in (XS a }

in

S , then

is a direct summand of

THEOREM 12.2.

Assume that

R[S] R

R[S]

, so

is a free basis for

is cancellative. If H

is direct.

saQ

R[S]

is the repre­

R[H]X a°

=

R[H] ,

.

is a unitary ring and

149

that

T

is a unitary extension ring of

R .

Consider the

following conditions. (1)

R

is an algebraic retract of

(2)

R

is a direct summand of

(3)

Each ideal of

(4)

T

(5)

If

R

T

T . as anR—module.

is contracted from

meets the total quotient ring of T

T . R

in

R .

satisfies the ascending chain condition

principal ideals, then so does

on

R .

(6)

If

T is a GCI>-domain, then so is

(7)

If

T is a factorial domain,

R.

thenso

is

R

.

The following diagram indicates implications that exist among these seven conditions.

(6)

I I (7)

(1)

— - (2) —

(4)

I (5)

Proof.

If

R

exists an ideal B n R = (0) and

(3) —

(1)

.

B

of

T

such that

In particular,

implies

T = R © W , where an ideal of

is an algebraic retract of

B

W

R , then

is an

T = R + B , where

is an

(2) .To see that

T , then there

(2)

R— submodule of implies

R— submodule of

T .

T

(3) , let If

B

is

BT = BR © BW = B @ BW , and hence

BT n R = B . (3)

= > (4)

contracted from R

and let

and T .

(5) : Assume that each ideal of Let

U

R

is

be the total quotient ring

of

t = a/b e T n U , where

regular in

R .

Then

a

regular in

U , it follows

e

and

b

is

bT n R = bR , and since

b

is

that

a,b

€ R

t = a/b c R .

Thus

(4)

is

150 satisfied.

Moreover,

principal ideals of in

T . If T

for some

k,

(1) = >

if

R ,

satisfies

a,b

(6) :

e R

<J>(c)R = aR

then

is an ascending sequence of

{a^T}

a.c.c.p.

is an ascending sequence

, then

and upon contracting to Assume that

algebraic retract of Take

{a^R}“

T

and let

R

and assume that cT = at

n bR . We have

(c)R = aR n bR , and (1) = >

(7) :

we have

4> :T — ► R

n bT .

a^R = a^+^R= i.. .

defines T

for

is a

R

as an

GCD—domain.

We show that

(c)R = <j>(cT) = <j>(aT n bT)

<J>(aT) n (j)(bT) = aR n bR = 4>(aR n bR) fore

a^T = a^+1T = ...

R

c

CaT n bT)

is also

a

.

c

There­

GCD— domain.

Factoriality of an integral domain

J

is

defined solely in terms of the multiplicative semigroup of nonzero elements of J

J , and hence Theorem 6.7 implies that

is factorial if and only if

a.c.c.p.

is satisfied in

it implies both

(5)

J .

J

is a GCD— domain and

Thus

(1)

implies

(7) since

and . (6) .

Theorems 12.1 and 12.2 cover the applications we need, save one.

That application concerns the case where

cancellative monoid and 0 .

H

is a subgroup of

S

S

is a

containing

Theorem 12.3 places the result needed in a general con­

text; we note that the extension

R[H] £ R[S]

satisfies the

hypothesis of (12.3). THEOREM 12.3. the unitary ring basis (B } A

R

T

is a unitary extension of

and that as an

{1} u (ta }ae^ , where each

A€ A

We have

R—module, ta

is a family of ideals of

Proof. hence

Assume that

A

R , then

®

has a free

is regular in

B T = B 0 (E © B t ) A

T

A

^

T .

If

(nB )T = n(B T) . A

for each

A

X

, and

151

<\'V> ■ W Because each each

a .

ta

T

T

0 .

An element

if ntc S t e T

containing

in

T .

If

for some

Sq

S

S

be a submonoid of

S

;S Q

S =

is said to be integral

n e

Z+ . The set SQ

S q , we say SQ

S

be integrally closed if

s e S

the case where

t eT such

of all

is integrally closed in T

S = SQ .

S

S

and

S

In defining

quotient group of

S . In

is almost integral over

that

s + nt e S

for each

S

monoid of

T containing S , and

closed in

T if

S = S*

.

completely integrally closed

S = S* .

S , then

S*

S , and If

integral over

T .

T .

In S ,

is said to the concepts

is cancellative and

the complete integral closure of

closure of

S

is the quotient group of

n e Z+ .

t eT S

The set

S*

is called

it is a sub­

is completely integrally

If

T

S*

need not be

is the

is called the complete

S

quotient integral

is completely integrally closed if

is integral .

T ;

S

In general, in T .

is a

Sif there exists

rn S

T

this case, an

t e T that are almost integral over

group of

of

to integrality for almost integrality, we r e ­

submonoid of the element

is a submonoid

is integrally closed in

is called the integral closure of

strict to

S

of

is called the integral closure of

is cancellative and

corresponding

T

t e T

that are integral over

It is easy to see that case

^or

® V nBX^a ■ W BX)T •

be a monoid and let

containing

elements

= ^nABA^ ta

Consequently,

Let

S

■

is regular in T ,

"A = W

over

•

over

S, then

t

The next result establishes the

is almost converse

152 under appropriate hypotheses on THEOREM 12.4. be

a submonoid of (1)

t

group of

Xt e R[T]

If

satisfies

and

T .

be a cancellative monoid, R

be a unitary

is integral

over

let

S

ring.

R[S] if and only

if

S .

Assume that S .

T

T , and let

is integral over (2)

S

Let

S

Tis a submonoid of the

t eT

a.c.c.

quotient

is almost integral over

for ideals, then

t

S

and

if

is integral over

S . (3)

If

R

is an integral domain and

of the quotient group of of

q.f.(R[S])

R[S]

Proof. X*

(l):

Xt

is almost integral over

is almost integral over

We show that

is integral over

let

R[S]

fore

for some

s e S

for some

is cancellative.

S

if

; the converse is clear. Thus,

nt e Supp(h^Xlt)

nt = s + it

S .

t is integral over

Xnt + h 1X (n’1)t+ ...+h n = 0 ,where each n-1 0 7

It follows that

T

t

is a submonoid

S , then considered as an element

, an element

if and only if

T

This proves

and

that

h. £ R[S] l

i < n .

(n - i)t = s t

There­ since

is integral over

S . (2):

Let

s e S

be such that

+

n e Z

.

The ideal

{s +

un = itCs + nt) + S] of

for some

for some

(3): if

m .

n < m + 1 .

and therefore

R[S]

s + nt e S

for each

°o

t

Hence

the converse.

s + ( m + l ) t £ S + n t + S

It follows that

(m + 1

- n)t £ S ,

is integral over S .

The assertion that t

S is generated by

Xt

is almost integral

is almost integral over Thus,

let

f £ R[S]\C0)

R

is clear. be such that

over

We prove

.

153

fXnt e R[S] that

for each

s + nt e S

integral over

for all

R

element of

s e Suppff)

If

n , and therefore

Assume that

is a subring of T .

Let

U

T

T

t

is a unitary integral d o ­

be a torsion— free cancellative

S

g e T[U]\(0)

has canonical form

be a submonoid of

U . Assume

g =

and each

g^ u^

that .Then

belongs to the complete integral closure of

R

is almost

containing the identity

monoid and let

only if each

, it follows

S .

THEOREM 12.5. main and

n e Z+ ,

R[S]

g

if and

belongs to the complete integral closure of belongs to the complete integral closure of

S .

Proof.

Denote by

integral closures of g. € R* si

and

si e S each

(R[S])* , R* , and R [S] , R , and

u. e S* 1

for each

are chosen so that k e Z+ , then let Clearly

each

, and hence

k

e Z+

Conversely, fgn e R[S]

order on write

U

if

rXs

g

S .

Assume first that

i .

If

r. e R\C0) l

€ R

and

+

g e (R[S])*

c (R[S])*

for each

.

is

f^X5!

Therefore

This implies that belongs to

for

Let

^

be such

be a total

compatible with the semigroup operation, and

+ k e Z

s =

f e R[S]\{o)

f, * 0, s, < s~ <...< s^ , and, 1 ’ 1 2 n

without loss of generality, u^ < u 2 < . . . fg^

f°r

.

, let

n e Z+ .

and €s

•g^ = rXs (E^g^X1^ )^ eR[S]

f = E? nf.Xsi , where i=l i *

term of

the complete

r = r,r~...r and let 1 l m

si+ ..-+sm •

that

k ^g^

S*

(R[S])*

< um .

• Cg1XUl)^ = f1g^XSl+ ^U1

k f^g^ 6 ^ g^ e R* n T[U]

and

and .

u^

s i + ^u i 6 ^ e S* .

By induction,

Hence

The initial for each ^or a ^

^ *

g - g^Xul

it follows that

154 g.

6i

e R*

and

u. e S*

for e a c h

i

COROLLARY 12.6. and let

G

(1)

Let

D

i .

be a unitary integral domain

be a torsion— free group. D[G]

is integrally closed if and only if

D

is

integrally closed. (2) D

D[G]

is completely integrally closed if and only if

is completely integrally closed. Proof.

Let

K

be the quotient field of

D[G] n K = D , it follows that

D

(completely integrally closed)

if

perty.

Assume that

y e q .f .(D[G ])

D

finitely generated subgroup and

y

is integrally closed D[G]

has the same p ro­

H

D [G ] . of

G

There exists a

such that

is integral over

D[H]

a finitely generated torsion— free group, n , and

D[H] -

Since

is integrally closed and that

is integral over

y e q.f.(D[H])

D .

H

.

Since

s Zn

H

is

for some

, . . . ,Xn ] [X'1 , . . .X^1 ] , a quotient ring of

the polynomial ring

D[X^,...,Xn ] .

Since the class of inte­

grally closed domains is closed under polynomial ring forma­ tion and under quotient ring formation, is integrally closed.

Therefore

it follows that

y e D[H]

c D[G]

D[H]

, and

D[G]

is integrally closed, as asserted. To prove

(2) , we note that

grally closed in to prove that

K[G]

K[G]

D[G]

by Theorem 12.5.

is completely inte­ Hence,

it suffices

is completely integrally closed.

If

{H^}

is the family of finitely generated subgroups of

then

K[G]

subrings of closed by

is the union of the directed set K[G]

.

Each

K[HJ

{K[HJ }

G , of

is Noetherian and integrally

(1) , hence completely integrally closed.

It

155 follows that

K[G]

is also completely integrally closed, and

this completes the proof of Corollary 12.6. COROLLARY 12.7. and let

S

Let

D

be a unitary integral domain

be a torsion— free cancellative monoid with q u o ­

tient group

G .

(1)

D[S]

(2)

D

Proof.

The following conditions are equivalent.

is completely integrally closed.

and

S

are completely integrally closed.

Theorem 12.5 shows that and

S

implies

Conversely,

if

then

is completely integrally closed in

D[S]

D

(1)

are completely integrally closed,

is completely integrally closed by part 12.6. (2)

Therefore

D [S ]

(2) .

(2)

D [G] , which of Corollary

is completely integrally closed if

is satisfied. In view of Corollary 12.7, it is natural to conjecture

that

D[S]

is integrally closed if and only if

are integrally closed.

D

and

S

This conjecture is established in

Corollary 12.11, but some preliminary work is required for the proof of (12.11).

Basically,

the link of the proof that

is missing at this point is the result that tegrally closed in G .

D[X;G]

if

S

D[X;S]

is in­

is integrally closed in

For this, we prove an analogue of Theorem 12.5; a couple

of new concepts are needed. Let

G

homomorphism say that

v

nontrivial.

be a group. v

G

G

The set

G , we mean a

into a totally ordered group

is trivial if

value monoid of group

of

By a valuation on

v(G) = {0 } ; otherwise,

Gy = (x e G |v(x)

>0}

v ; it is a submonoid of

and it is easy to see that

Gy

G

H . v

is

is called the with quotient

is integrally

We

156 closed.

The next result is the analogue of a well known re­

sult for integral domains. THEOREM 12.8. monoid of

G .

such that

G

in

be a group and let

S

be a sub-

be the set of valuations on

Then A g rt V u a

a

G

is the integral closure of

G . Proof.

If

G

Let {v } A a aeA

^ S . V

S

Let

Let

T

be the

t e T , then

t

is integral over each

t € AG a va

since each

integral closure of

S

in

G, and hence va is integrally closed in G .

G va

G .

This

proves that x

T cf)G . To prove the reverse inclusion, take « va Then x is not integral over T ,and our aim is

eG\T .

to show

that

x^ Gv

for some

invertible elements of

T .

G/H , as follows:

(2)

is a

not integral ment of hence

g

(1)

G/H

over

T/H .

Thus,

g

and

, -g

e T .

- g

if are

Therefore

torsion— free, as

asserted.Statement

definition of

, and (3)

H

T .

G/H , and

g+ H

<

t e T

x + H

a

on

G/H '

integral over g, (2)

such that

T

g € H follows x

<

.

0 < t + H a

be the Then

is

, and and

G/His

from the is not inte­

by choice of the ordering

£

2 T , and

.

and let

v

be a

a

Therefore f \G

E T .

a va

valuation on the

Vbe the value monoid

reduced unitary ring

G

x ^ G

va

a

THEOREM 12.9. Let

for each

canonical map of

G

a

G

x +H

is a torsion e le ­

is true because

< 0 . Let v a

ontoG/H , ordered under

group

(3)

Corollary 3.5 implies that there exists a

total order and

H be the group of

is tors ion— free,

positive— subset of

G/H ,then

gral over

Let

We wish to prove three assertions

concerning T/H

a .

R , the monoid ring

of R[V]

v .

torsion— free For any

is integrally

157 closed in

R[G]

Proof. element

.

Assume,

f

of

to the contrary,

R[G]\R[V]

that there exists an

that is integral over

Without loss of generality, we assume that each

s e Supp(f)

in the form where each

.

If

H = ker(v)

z!?_iXsih^ , where Ik

R[V]

v(s) < 0

, then

f

vCs^) < ...

for

can be written

< v(sm ) < 0

is a nonzero element of R[H]

.

and

.We assume

that f + a n where R[G]

a^ e R[V]

, fn ^+... + a,f n-1 1

for

each

i.

Since

is reduced by Theorem 9.9.

Each element of

Supp(XnSlhi)

G

is tors ion— free,

has

v(s)

implies for each

must belong to for some

s

> nvfs^)

we obtain

Now

the form

, E^k^= n , k^ < n , and v(s)

for

= E^k^vCs^)

, where

h e

H .Applying

and

c e SuppCf1) .

the argument just given shows that

nv(s^)

e

fn + Z q ^a^fn ^ * 0 .

k^

> 0 v ,

> E^k^vCs^) = ( ^ k ^ )vfs^) = nvfs^) .

where w e Supp(a^)

v(s) = v(w)

or to

+ h

s e SuppCa^f1)implies

proved that

To o b ­

s c Supp(fn - X ^ l h ^ )

On the other hand,

quently,

.

•

Supp(fn - X ^ l h ^ )

i < n .

is of j

nvfs^)

, where

s

Supp(a^f^)

Xnslhi * 0 .

v— value

g = ( f n - xnsi h ? ) + Each such

0,

Therefore

tain a contradiction, we show that s e Supp(g)

+ an = 0

v(c)

s Thus

= w + c v(w)

> ivCs^) .

+ v(c) > ivCs^) > n v C s ^ Supp(fn + Z^ ^a^fn

^ 0 , and Conse­

. We have

, and hence

This completes the proof.

,

158 The analogues of Theorem 12.5 and Corollary 12.7 follow easily from the two preceding results. THEOREM 12.10. R

is a subring of

Let

U

(1) R

is a unitary domain and

containing the identity element of

T .

S be a

U .

R[S]

is integrally closed in

is integrally closed in T and

S

T[U]

if and only if

is integrally closed

U . (2)

The integral

R ’[Sf] , where S'

R'

closure of

is

R[S]

(1) :

is clear that

If

R[S]

R = R1

Since

U

is R

in

in T

S = S'

and

U .

is integrally closed

and

verse, assume first that Sf =A u r a va

in T[U]

the integral closure of

is the integral closure of S Proof.

it

T

T

be a torsion— free cancellative monoid and let

submonoid of

in

Assume that

.

is a group.

in T[U]

To prove the

,

con­

By Theorem 12.8,

is an intersection of valuation monoids on

U .

] is integrally closed in T[U] for each a by a Theorem 12.9, it follows that the integral closure of R [ S ] in

T[U

T[U]

is contained in A T [ U ] = T[nU ] = T [ U ’] . To a va a va complete the proof in the case where U is a group, it therefore suffices to prove that closed in

T [ U f] .

showing that

is integrally

Simplifying the notation,

R[S]

is integrally closed in

integrally closed in f e T[S]

R ’[U1]

T

integral over

.

this amounts to T[S]

if

R

To prove this, take an element

R[S]

.

If

G

is the quotient group

of

S , then there exists a finitely generated subgroup

of

G

such that

f e T[H]

is

and

f

is integral over

As in the proof of Corollary 12.6, however,

H ^ Zn

H

R[H]

.

for some

159 n .

It is known that R

that

R[X^,...Xn ]

hence

R[H]

integrally closed in

is integrally closed in

~ R[Zn ] ~ R[X ^,...»Xn ]g

each coefficient of

f

pletes the proof of

(1)

If

U

. is in

R[S]

closure of

S

R[S]

is the multiplica­

Therefore

in

f e R[S]

let

is

W .

in T[U]

f e

W

U

R[H]

R 1[M]

This com­

is

a group.

be the quotient groupof

,where

Therefore, is

R'[M]

M

is the in­

the integral

n T[U] = R'[M n U] =

R' [S' ] . Statement

(2)

COROLLARY 12.11. and let

S

Cl) D*

D[S]

D

is

D'[S']

, where

and S'is the integral

is integrally closed if and only if

D

and

are integrally closed. Proof. (2)

We need only

of Theorem

gral closure of group of

that

D [S ]

S .

S

D ’[G]

be a unitary integral domain

The integral closure of

is the integral closure of

(2)

(1) .

be a torsion— free cancellative monoid.

closure of

Part

of (12.10) follows easily from Let D

S .

D[S] Since

prove (1) ,

for

12.10shows that

C2)

then follows.

D f[S']

is the inte­

inD 1[G] , where

G

is

the quotient

D 1[G] is an overring of

D

and since

is integrally closed by Corollary 12.6, we conclude D'[ S’]

is the integral closure of

,

.

implies that the integral

in T[W]

tegral closure of

B

R , and

we

U .The case just considered closure of

T[X^,...Xn ] , and

in the case where

is not a group,

implies

is integrally closed in

T [ X 1 ,...,Xn ] g ~ T [ Z n ] ~ T [H] , where tive system generated by

T

D[S]

.

Having settled the problems of determining conditions

160 under which a monoid domain

D[S]

is integrally closed or

completely integrally closed, we conclude this section with a description of the integral closure of THEOREM 12.12. cal

N

and let

Assume that R

S

R

f

Then

tegral over

f

is integral over

is nilpotent for each R

s * 0 .

in

R[S]

It is clear that each element of

integral over

R . R .

.

be a torsion— free cancellative monoid.

is the integral closure of

Proof.

in R[S]

be a unitary ring with nilradi­

f = ^ses^s^S 6 R [S] •

if and only if

R + N[S]

Let

R

g = f - fQ

.

R + N[S]

For the converse, assume that

Then

Thus,

f

is

is in­

is also integral over

R .

Let

(*)

gn + r n- i « n" 1+• • -+r i g + r o =

0

be an equation of integral dependence for

g

over

P

<J>

is the canon­

is a proper prime ideal of

ical homomorphism of

R[S]

R , and if

onto

(R/P)[S ] , then

R .

(*)

transformed to an equation of integral dependence for over

(R/P) [S] .

It follows that

take a total order deg 0

or

£

on

S .

ord
Then by choice of

that

g e N[S]

passing that 9.9.

g e P[S]

.

Since

g , either

deg

P

Therefore

is arbitrary,

, and this completes the proof. N[S]


If, for example,

deg((j> (r^_ ^gn 1+ ... + g 0)) , a contradiction. and

is


deg (Kg) > 0 , then we conclude that

4>(g) = 0

If

is the nilradical of

R[S]

it follows

We note in by Theorem

161

Section 12 Remarks For polynomial rings, Theorems 12.5 and 12.10 and Corollaries 12.6 and 12.11 are known of [51]).

(see Sections 10 and 13

But for rings with zero divisors, equivalent con­

ditions for the polynomial ring to be integrally closed or completely integrally closed are not known.

If

R [X ]

is

integrally closed, then

R

is integrally closed and reduced,

but the converse fails.

For more on this topic, see [3].

Power series rings over integral domains behave like monoid domains in regard to the properties of complete inte­ gral closure covered by Theorem 12.5 and Corollary 12.6. But for integral closure, the situation is quite different. For example, for fields

F

and

K

with

K [[X]]

need not be integral over

D [ [X]]

need not be integrally closed if

closed.

Of*course,

closed, then grally closed.

if

D

D , and hence

K/F

F [ [X]] . D

algebraic, Moreover, is integrally

is Noetherian and integrally D [[X]] , are completely inte­

Again Section 13 of [51] and its references

provide additional information on this topic.

162 §13.

Monoid Domains as Prufer Domains

The closely related classes of Prufer domains, Bezout domains, Dedekind domains, and principal ideal domains are of fundamental importance in commutative ring theory.

In this

section we determine necessary and sufficient conditions in order that a given monoid domain of the four classes named. Section 13 that that

S

D

D[S]

should belong to one

Thus, we assume throughout

denotes a unitary integral domain and

is a torsion— free cancellative monoid.

We begin

with the problem of determining conditions under which

D[S]

is a Prufer domain. THEOREM 13.1. D

Assume that

and that each ideal of D

is a Prufer domain, then Proof.

is contracted from

the quotient field of

D

is contracted from

D .

J ,it

conclude that

(a,b)

COROLLARY 13.2. If

H

is

Prufer domain.

J

D .Let

K

J n K = D , and

is a Prufer

domain,

D ,we

as we wished to show. D[S]

containing

In particular,

a,b e

Contracting to

2

Assume that S

If

is integrally

for all

Since J

= (a ,b )

a subgroup of

D

follows that

(a,b)2J = [(a,b)J]2 =(a2 ,b2)J . 2

J.

Since each principal ideal of

is integrally closed.

2

domain of

is a Prufer domain.

(a,b)2 = (a2 ,b2)

be

D

is an extension

It suffices to show that

closed and that

hence

D

J

is a Prufer domain. 0 , then

D = D [{0>]

D[H] is

a

is a Prufer

domain. Proof.

Theorem 12.2 shows that each ideal of

contracted from

D[S]

.

D[H]

is

163 THEOREM 13.3.

If

domain, then either Proof. such that

D

s

is not

invertible in

S

i

s % Suppfx

ficient since

f^

S

g)

and if

D [S] S

invertible.

is a group.

d e D\{0)

If

.

S

s

is not

is cancellative, it 2

Therefore

d = d f^

We conclude that

d

D

e s

s

, then

dXs = d 2f + X 2sg . Since

f , and consequently,

1 = df^ .

is a Priifer

is not a group and choose

and since

2s

of

0

is a field or

Assume that

dXs e (d2 ,X2s) , say

that

S *

follows

for some coef­ is a unit of

is a field if

S

D is

not a group. Recall that a group nx = g

is solvable in

An arbitrary group group

H

phism.

H* Thus

can

is divisible

for each

g e G

If

H

H*

and each

G .

is uniquely determined

Hp , where

is a vector

n e Z+ .

As is implied by up to isomor­

istorsion—free, then considering H

Z , each nonzero element of is merely

if the equation

be imbedded in a minimal divisible

H* called the divisible hull of

the terminology, H*

over

G

G

T

is regular on

H , and

is the set of nonzero

integers.

space over

Z

as a module

Q , and as a group is isomor­

phic to a direct sum of copies of the additive group

Q .

The statement of Theorem 13.4 involves the divisible

hull.

THEOREM 13.4.

Assume that

group with divisible hull H* .

H

is a nonzero torsion— free

The following conditions are

equivalent. (1)

D[H]

is a Priifer domain.

(2)

D[H*]

(3)

D

is

is a Priifer domain. a field and

H* = Q .

164 (4)

D[H]

Proof. implies

is a Bezout domain.

(1) < = > (2) :

(1) .

For the converse, assume that

Priifer domain. Then

Hm

For

m e Z+ , let

is a subgroup of

H = u00 H m= 1 m D[Hm ]

Corollary 13.2 shows that (2)

, where

H*

D[H]

is

Hm = {g e H*|(m!)g

isomorphic to

H, c H 0 c . . . . 1 “ 2“

H

a

e H) .

and

It follows that each

is a Priifer domain, and hence D[H] = u~D[Hm ]

is also

a Priifer domain. (2) G

of

D[Q]

=>

H* . is

(3) : Then

We write

H*

as

D[H*] a D[0][G]

a Priiferdomain.

.

Since

valent,

D [Z ]

a field,

take a nonzero element

Q © G

for some subgroup

By Corollary 13.2,

(2)

is also a Priifer domain. d e D

and (1)

are equi­

To show that and let

D

is

A =

(d2 ,(l - X ) 2) be the ideal of D[X] generated by d2 and ? (1 - X) . Since A and XD[X]are comaximal, it follows that

A

is

contracted from

have

d(l - X) e AD[Z]

f

and

g

unit of

D .

is a Priifer domain.

f,g

Consequently, that

some

Consequently

= A .

Write

t D[X]= D[1 - X ] 1 - X

d e d D , and this

we conclude G^ .

D[Z]

n D[X]

as polynomials in

it follows that

= D[X,X *] = D[Z]

since

Therefore d(l - X) e AD[Z] d 2f + (1 - X ) 2g , where

D [ X ] ^

d(l - X)

implies that

D[Q] .

Therefore

proof that

(2)

1 - X

d

D,

is a

Returning to

G = {0} , for if not, then

is not a field because

=

Regarding

G = Q © G^

H* , for

is a Priifer domain, and

this is a contradiction to what was just proved — D[ Q ]

We

with coefficients in

Dis a field.

D[Q @ Q]

.

,

to wit,

is a nonzero nonunit of

G = {0} , H « 0 , and this completes the implies

(3) .

165 (3) by

=>

(4) :If Gm

1/m! for each

u?G. 1 m

Therefore

m > 0 , then

and

D D[Q]

and

generated and

D[G ] ~ m

D [ X ], a principal

is a field.

of Q

£ G2 E •••

D[Q] = u"d [G ] 1 m

a quotient ring of since

is the subgroup

Therefore each

Q =

DfX.X'1 ]

ideal domain

(PID)

D[G ] m

PID ,

is a

is

is a Bezout domain.

(4)

implies

(1) because a Bezout domain

is a Prufer

domain. If then

G

is any abelian group —

G is a

torsion— free or not —

Z—module and we can consider the group

a vector space over the field

Q =

, where

G^, as

T = Z \{0> .

The dimension of this vector space is called the torsion— free rank

of G

and is denoted by

ro(G)

>

charac-

terized as the supremum of the cardinalities of free (that is, linearly independent) subsets of and only if group

G

is

is a subgroup of

rank of

r o(G)

We use

If

D[S]

of

S , then

r^fM)

that

D is a field and

G

M .

is the quotient group D[S]

, and hence

also a Prufer domain.ByTheorem 13.4, it follows

Moreover, that either

if

rQ (S) S

S £Q q

=1

if

D[S]

is

a Prufer

is not agroup, then Theorem 2.9 or

- S

set of nonnegative rational numbers. then

.

is the quotient group of

is a quotient ring of

is

shows

> where G

Q

the torsion— free

to denote the torsion— free rank of

D[G]

domain.

H

wedefine

is a Prufer domain and if D[G]

^

if and only if semigroup M ,

M .

ro(G) = 0

for a torsion— free

For a cancellative to be

Thus

a torsion group, and

H , rQ (H) = 1

M

G .

£ Q q , where Q q Thus, if

S is isomorphic to a submonoid of

Qq

is the

S * {0}

containing

, 1 .

The next result determines conditions under which the monoid

166 ring of such an

S

THEOREM 13.5. is a submonoid of tient group of

, over a field F ,

is a Prufer domain.

Assume that

a field and that

Qq

S .

Fis

containing 1 .

Let

G

S

be the quo­

The following conditions are equivalent.

(1)

S

is integrally closed.

(2)

G n Q0 - S .

(3)

S

is the union of an ascending sequence of cyclic

monoids. (4)

F[S]

is a Bezout

domain.

(5)

F[S]

is a Prufer

domain.

Proof.

(1) = >

over

Zq £ s .

over

S , and

(2) :

Each element of

Hence the elements of G nQ q = S

(2) =“ > (3) :

since

Part

(1)

S

Qq

G n Qq

is integral are integral

is integrally

closed.

of Corollary 2.8 shows that

is the union of a chain of cyclic groups

{ ( g ^ ) •

out loss of generality, we assume that

> 0

Then

S = G n q q = [u"(g-)]

where

- ^ i +l ^ (3) “ > (4) :

" Q0 =

^or each

G

With ­

for each

i .

n Q01 = ^ i *

’

i •

It follows from

(3)

that

FI S]

is the

union of a chain of subrings, each isomorphic to F [Zq ] = F[X] , a That

(4)

PID . implies

is a Prufer domain, then S

Therefore (5)

F[S]

is clear.

F[S]

is a Bezout domain. Moreover,

if

F[S]

is integrally closed, so that

is integrally closed by Corollary 12.11. We note that condition

trinsic property of

S

that

assumptions that S c Q q submonoid

T

of

Q

and

(3)

of Theorem 13.5 is an in­

requires noreference to that

a Prlifer monoid

1e S

. Wecall a

if T

the nonzero

is the union

167 of an ascending sequence of cyclic submonoids.Theorem is a summary of the results of this

section up

the hypotheses of Section 13 concerning

D

13.6

to thispoint;

and

S

are re­

peated in its statement. THEOREM 13.6. main and that monoid.

S

Assume that

D

is a unitary integral do ­

is a nonzero torsion— free cancellative

The following conditions are equivalent.

Cl)

D[S] is a Priifer domain.

C2)

D[S]

C3)

D

is a Bezout domain.

is a field and to within isomorphism,

either a subgroup or a Priifer submonoid of

S

is

Q .

A Dedekind domain can be characterized as a Noetherian Priifer domain, and a

PID

is a Noetherian Bezout domain.

In view of these facts, it is easy to determine from Theorem 13.6 conditions under which

D[S]

aPID

proof of (13.8) uses a basic

CTheorem 13.8).

The

is a Dedekind

domain or

result that is also referred to in Sections 14 and 15. THEOREM 13.7. contains in

If

<s>° , then

D[S] .

Thus,

if

s,t e S (1 - X1*) D[S]

are such that ^ properly contains

satisfies

condition for principal ideals

properly (1 - X s )

the ascending chain

(a.c.c.p.)

, then

S

satis­

fies the ascending chain condition on cyclic submonoids. Proof. 1

We have s = nt

- Xs = (1 - X t)(l +

nonunit of

DIS]

properly contains

for some

n > 1 .

X t+...+ X (n‘1)t)

by Theorem 11.1. (1 - X s) .

Therefore ,

Consequently,

where is a (1 - X t)

168 T H E O R E M 13.8. Theorem

13.6.

The

and

(2)

D[S]

or to

is a

D[S]

S

be

as

following conditions

D [S ] is a Euclidean

(4)

in the s t a t e m e n t

of

are e q u i v a l e n t .

domain.

PID .

is a Dedekind domain.

D is a field and

S

is isomorphic either

to

Z

Zq . Proof.

=>

(4)

We need only establish the implications and

domain, then

D

a submonoid of is Noetherian that

D

(1)

(3)

(3)

Let

S

(4)

=>

is a

(1) .

Thus, if D[S]

field by Theorem 13.6.Moreover,

0 , and

S

If

that is,

S

is a group,

S - Z .

group, then Theorem 13.5 shows that

If

S

Consequently, shows that (4)

satisfies

a.c.c.

S = <s>^ for some (3)

implies

— > (1) :

If

a Euclidean domain.

(4)

D

S

is

D[S]

it follows is not a

is the union of an

ascending sequence of cyclic submonoids. S

S

is finitely generated since

(Theorem 7.7).

is cyclic —

implies that

isa Dedekind

But Theorem 13.7

on cyclic submonoids. s , and

S ^ Zq .

This

.

is a field,

thend [z q]

= D[X]

is

Since a quotient ring of a Euclidean do­

main is again Euclidean,

it follows that

D[Z]

is also

Euclidean. Section 13 Remarks A brief account of the theory of Priifer domains is given in Section 1— 6 of [83]. Chapter IV of [51]. shows that the domain D

For a more detailed account,

see

In particular, Theorem 24.3 of [51] D

is a Priifer domain if and only if

is integrally closed and

(a,b)

2

2

2

= (a ,b )

for all

169 a,b e D ; we have used this characterization of Prufer d o­ mains several times in Section 13. Our restriction to integral domains in Section 13 will result in some duplication of results in Section 18, where the problems of determining conditions under which

R [S ]

a Prufer ring, a Bezout ring, etc.

On the

are considered.

is

other hand, a more complete treatment of ring— theoretic aspects of the problem are included in Section 18, for the classes of Prufer and related rings are not as familiar as their domain counterparts. Another class of domains frequently considered in con­ nection with the classes of Prufer domains and Dedekind d o ­ mains is the class of almost Dedekind domains, defined as follows. main

An almost Dedekind domain is a unitary integral d o ­

D

such that

Dp

each proper prime ideal

is a Noetherian valuation domain for P

of

D .

Dedekind domains are al­

most Dedekind and an almost Dedekind domain is a Prufer domain.

Except for the class of fields, almost Dedekind d o ­

mains are one— dimensional. domain

D

is Noetherian

A one— dimensional almost Dedekind

(hence a Dedekind domain) if and

only if no maximal ideal of

D

is idempotent.

Another char­

acterization is that the almost Dedekind domains are the domains in which the cancellation law for ideals holds — is,

AB = AC

and

A * (0)

implies

B = C .

that

Gilmer and

Parker in [59] determine necessary and sufficient conditions for

D[S]

to be an almost Dedekind domain, but their proof

requires specialized results about such domains that are out­ side the scope of this monograph.

In Corollary 20.15,

however, we determine those group rings

D[G]

that are

170 almost Dedekind. D[G]

That result is strong enough to show that

need not be Noetherian if it is almost Dedekind;

example,

Q[Q]

for

is such a domain.

For a brief account of the topic of divisible abelian groups, the interested reader should consult Chapter IV of [47].

171 §14.

Monoid Domains as Factorial Domains

As in Section 13, we assume throughout this D

is a unitary integral domain and that

cancellative monoid. monoid domain

is factorial.

factorial if and only if J

is a torsion— free

We seek conditions under which the

D[S]

is satisfied in

S

section that

.

J

A unitary domain

is a GCD— domain and

J

is

a.c.c.p.

Therefore one approach to the problem

of determining conditions under which

D[S]

is factorial

would be to determine separately conditions under which is a

GCD— domain or satisfies

results together.

a.c.c.p.

This is close to

The hedging is on the

a.c.c.p.

,then mesh

D[S]

these two

the approach we

follow.

part, but the first step in

the proof is to determine necessary and sufficient conditions in order that

D[S]

condition is that operation on

should be a S

should be a

GCD— domain.One necessary GCD—monoid.

S is written as addition, a

the notation

of Section 6 is required;

valent form of the definition of a additive notation, u e s

such that

Since the translation from

for example, one equi­

GCD—monoid, using

is that for all

s,t e

S , thereexists

(s + S) n (t + S)

= u+

S .

The first two

results come easily. THEOREM 14.1. GCD— domain and Proof.

S

If

D[S]

is a

Since

D

is a

is an algebraic retract of

that

D

S

take

s,t € S

such that

GCD-monoid, (Xs)

divisible by

D

is a

GCD—monoid.

Theorem 12.2 shows is a

GCD—domain, then

n (x*) = gD[S] g , part (1)

is a

D[S]

,

GCD—domain. To prove that and let

g e D[S]

. Since the monomial

of Theorem 11.1 shows that

be X s+t g

is

iii is a monomial, necessarily of the form unit of

D .

Thus

(Xs) n (X1) = (Xw ) .

v e S , is the semigroup ring (s + S)

uX

D[v + S]

, where Since

is a

(Xv ) , for

, it follows that

n (t + S) = w + S .

THEOREM 14.2.

If

torsion— free group, Proof.

Before

quotient ring

JT

D is a

then

GCD— domain and G

D[G]

of a

GCD— domain is again a

aJ n bJ = cJ

it follows that

a

giving the proof, we observe that each

aJT

and

Without loss of generality we assume that Then

is

is a GCD— domain.

To prove this, take principal ideals

J .

u

for some

a

c c J .

GCD-domain. bJT and

of b

JT .

are in

Extending to

aJT n bJT = (aJ n bJ)JT = cJT , so

JT

, is a

GCD-domain. To prove that ments

f = E^f\Xsi

D[G]

is a

GCD-domain, take nonzero

, g = Ejg^X^

of D[G]

, and let

e le­

H

be

the subgroup of G generated by the set {s ^ , . . . ,s^t^, .. v Then H ^ Z" for some k , and D[H] is isomorphic to a quotient ring of the polymonial ring nomial ring over the

D[X^,...,X^]

GCD-domain is again a

result in the preceding paragraph,

GCD-domain. h e D[H]

.

Therefore

fD[H]

.

GCD-domain,

A poly­ and by

D [H ] is also a

n gD[H] = hD[H]

for some

It then follows from Theorems 12.1 and 12.3 that

fD[G] n gD[G] = hD[G]

.

This completes the proof of Theorem

14. 2. In Theorem 14.5, we establish the converse of Theorem 14.1.

The proof of (14.5) uses two preliminary results, plus

a couple of new concepts. D\ {0} , and if

x

If

A

is a nonempty subset of

is a nonzero element of

D , we say that

.,tm > .

173 x

is prime to

alently,

A

x

if

is prime to

THEOREM 14.3. and let N .

aD n xD

Let

A

= axD

for each a

if xd:a = xD

e A , equiv­

for each

a e A .

Nbe amultiplicative system

T be the set of elements of

D

in

D

that are prime to

Assume that the following two conditions are satisfied. (1)

Each pair of elements of

multiple in (2)

D can be expressed in theform

is a GCD— domain, then

Proof.

Since the elements of

follows that aGCD—domain, b =

tDXT n D N

,c = n 2 t 2

’ w ^ ere

Moreover,

n t 2D^

T

are prime to t eT .

n i»n 2 6 N

we show that

bD

*

n cD .

If

and

t

(n^) n (s) = (n^s)

Hence x e bD

s

e

t l ,t2

and

c T . To

T .Then

, so that

Since

ru|m

D .

complete

for

n cD ,we write n^|ms

for

is In fact,

the proof, then

i = 1,2, x

Let

as

ms ,

i = 1,2,

; consequently,

and

and n|m

.

the other hand, xDXTn D = sD s b D xl n cDXT n D = tD , N N N so that

t|s

.

ntD , as asserted.

It follows that

is

.

D^ and

t e t.DXT , l N

nt e n^t^D

D

€ T *

c D N = t 2D N

a principal ideal of

ncD = ntD .

n D =

To show

is a principal ideal of

tlDN n t2DN = tDN^or some

N ,it

and write

an(*

bD^ = t^DN

therefore the extension of

m £N

is a GCD—domain.

take nonzero elements b,c e D

We ha ve

where

D

= tD for each

n = l c m { n 1 ,n2 > .

ntD c bD

nt

n e N ,t e T.

If

t e

has a least common

D . Each element of

for some

N

nt|ms = x

and

bD

n cD =

On

174 Assume that Then

S

D

is a

GCD— domain and

S

is a GCD—monoid.

induces a natural grading on the domain

homogeneous elements of nomials

D[S]

dXs . To prove that

to apply Theorem 14.3 to homogeneous elements of

under this D[S]

S-grading are the m o ­

is a GCD-domain, we intend

D [S ] , where D[S] .

D[S] , and the

N

is the set of nonzero

The next result paves the way

for the application of (14.3) in the setting indicated. f = Z^a^Xsi is the canonical form of a nonzero element then let

d = gcd(a^}^

up to associates

in

vertible element of

and let D

S .

and

s= gcd(s^}^ ; d

If f e D[S],

is determined

s is determined to within an in­

We call

dXs

the homogeneous content

(h— content) of_ f ; it is determined up to a unit factor in D[S] .

If

f has

h— content

1 , we say that

ly primitive (h— primitive) . Clearly

f

is homogeneous -

f = dXsf* , where

f*

is

h—primitive. THEOREM 14.4. is a

GCD-monoid.

Assume that Pick

D

is a GCD-domain and that

a,b e D \{0} , s,t e S , and let

lcm{a,b} , u = lcm{s,t) . Assume that tive.

h-primi-

(a) n (b) = (c) , where parentheses denote the ideal

D[S] generated by the given element. (2)

(Xs) n (X1) = (Xu ) .

(3)

(a) n (Xs) = (aXs) .

(4)

(aXS) n (bX1) = (cXU ) .

(5)

fg

(6)

(aXs) n (£) = (aXsf) .

Proof. (2): D [(s

are

The following statements are valid. (1)

of

f,g e D[S]

c=

+ S)

is

h— primitive.

Statement (Xs ) n (Xt )

n (t + S)]

=

(1)

follows from Theorem

= D[s + S]

n D[t

+ S] =

D [u +S] = (XU ) .

12.3.

S

175 (3):

(a)

n (Xs) = aD[S]

n D [ s + I] = aD[s + I] =

a(Xs) = (aXs) . Statement

(4)

follows from

It follows from

(4)

geneous elements of multiple in

D[S]

conclude that dXw , where (3)

Applying part

d = gcd{a,b}

(4)

and

. We use these

let

< be a total order on

< s0 <

is

l

. .. < s n

h—primitive,

gcd{Y}

S

and let

and

fg . Let

and let

of

= gcd{w,gi> = 1 , then

ponent of

fg

not divisible by

gcd{w,fi>

* 1 .

Since

is h— primitive,

For

1 < i < n

r

w^ = 1 .

that

, we

D[S]

and

and

To prove that

fg °

r

Then

fails todivide some

for j

for

= 1,2,...,m .

w . , let

a previous case, and if

If

com­

Thus, assume that w. = Choose

gcd{w,f^,...,f^} . i ^ 2

minimal

is a nonunit homogeneous

therefore assume thatw

gcd{w,f^} = 1 .

If

is a homogeneous

w , and it suffices to show that

minimal so that

Thus,

g , where

f^ = a^X51

w = 1 . n

divide some homogeneous component Wi-i

exist

f = Z^a^X5*

of f

g^ = b^X^j

gcdCwjf^}

divisor of

part

it suffices to show that each nonzero, no n­

homogeneous component of

such that

. Thus,

andlcm{Y}

t, < t0 < . .. < t . 1 L m

w

f

and is equal to

observations in proving (5) .

unit homogeneous element

i = 1,2,...,n

D[S]

w = gcd{s,t}

g = E^bjX^ be the canonical forms i

homo­

of nonzero homogeneous elements of

D[S]

s

.

of Theorem 6.1, we

exists in

of (6.1) then implies that Y

(3)

has a homogeneous least common

gcd (aXs ,bX^ }

for each finite set

and

that each pair of nonzero

D[S]

.

(1) , (2) ,

of

fails to

fg . Replacing

divides

w

f^,...,^!

by and

gcd{w,g1 > = 1 , we're finished by gcd{w,gi> * 1 , we choose

gcd{w,g^,...,gj} = 1 .

Replacing

j > 2 w

by

176 gcd{w,gj,...,g j _p w

, it suffices to consider the case where

divides each of

gcd(w,gj } = 1 .

f1 ,...,fi_1 ,

The

and

(s^ + t^)— component of

gcd(w,fi> =

fg

has the

v < i

or

form figj + (a sum Since

w

of terms fy£y > where either

divides each such

it follows that of

fg .

w

To prove

g = aXsk =

fq .

Let

h— content of k*

and

q*

q^ • q*f , where and

q*f

are

(5) .

(6) , we show that if

(aXs) n (f) ,

where

gcd{w,f^gj}= 1 ,

(s^ + t^)— component

the proof of

element of

be the

anc* since

fails to divide the

This completes

(6):

fy gy

then

and

q^

h— primitive.

differ by a unit factor in

write k

k = k^k*

h— primitive.

aXsk^

is a nonzero

h— content of

q , and write are

g

g e (aXsf) .Thus,

k^ be the

y < j).

,let

and

q = q^q*

Then aXsk^

It follows that , so

f|k*

k* .

,

• k* =

are homogeneous and

D[S]

q^

k*

and

Hence

q*f g =

aXsk xk* e (aXsf) . THEOREM 14.5. GCD-monoid, Proof.

then

If

D

D[S]

is a

GCD-domain

and

S— grading on

be the set of nonzero homogeneous elements of be the set of elements of

D[S]

h— primitive elements of

f = ‘aXs f*

D[S]

T

(f):aXs = (f*) * (f)

a

so that

D[S] .

.Part

(6)

h— content f

, let Let

N .

We

are precisely the

h— primitive element of has nonunit

D[S]

that are prime to

show that the nonzero elements of

And if

is

is a GCD-domain.

Under the canonical

shows that each

S

of Theorem 14.4 D[S]

is in

T

.

aXs , then

is not prime to

N .

We

N T

177 note that conditions isfied for part

(4)

Moreover D

and

that

N

and

(1)

and

T ; that

(2) (2)

of Theorem 14.3 are sat­ is satisfied is clear, and

of Theorem 14.4 shows that D [S }^ ~ K[G]

G

, where

K

is a

is satisfied.

is the quotient field of

is the quotient group of

D[S] n

(1)

S .

GCD— domain, and hence

Theorem 14.2 shows D[S]

is a

GCD— d o ­

main by Theorem 14.3. We turn to the problem of determining conditions under which

D[S]

is factorial.

necessary conditions on lish.

As with the

D

and

S

concerning

(1)

If

satisfies ticular,

Let

a.c.c.p.

D

K

be the quotient field of D .

is satisfied in

for each subgroup

satisfies

If

satisfies

14.6.

a.c.c.p. D

(2)

and

H

K [S ]

satisfy

S .

a.c.c.p.

Statement

(1)

In pa r ­

, then D[S]

follows from Theorem 12.2.

(2) , take an ascending sequence

i .

of

a.c.c.p.

of nonzero principal ideals of Since

K[S]

g^ is a unit of

trivial (Theorem 11.1), d^Xsi , where

D [S ] —

satisfies

out lossof generality that

form

D [S ] , then D[H]

a.c.c.p.

Proof.

each

We first prove a result

a.c.c.p.

THEOREM

each

are fairly easy to estab­

Theorem 14.7 represents an expanded analogue, for

factorial domains, of Theorem 14.1.

prove

GCD—property, some

Cf^) say

a.c.c.p.

£ (f2^ £ • • • f^ = g^f^+i

.

Since units of

it follows that each

s^ is invertible in

S

for

, we assumewith­

f^K[S] = f2K[S] = ...

K[S]

To

g^ .

.

Thus,

K [S ] are is of the Let

a^

be a

178

nonzero coefficient of some coefficient

a2 ^ 0T

f^ = d^X5! ^ of

f2 •

a2 =

have

a^D c a 2D £ ... , and since

some coefficient

D , it follows that

d^Xsi

is a unit of

...

This completes the proof of

.

THEOREM 14.7. let

G

then

D[S]

Let

for

K

and

S

for some

D .

k .

Conse­

Therefore (f^)

=

=

be the quotient field of

Moreover,

We

=

(2).

S .

are factorial and

cyclic submonoids.

f^ , etc.

is satisfied in

i > k ,and

be the quotient group of D

of

a.c.c.p.

are units of

^ 0T

f2 = d 2X S2fj ,

a^

a^D = a^+1D = ...

d^,d^+ i ,...

= ^ia2

Then

And since

then

quently,

.

If

S

D [S ]

and

is factorial,

satisfies

D[G]

D

and K[S]

a.c.c.

on

are factorial

domains. Proof. Theorem 12.2.

Since

D[S]

is factorial,

To prove that

invertible element

S

s e S .

Each

JL

where Then of

u. l

D

and p. *i

X s as a , say

take a non-

finite product Xs

= - --IIn .

of the form

u^X^ 1 ,

is not invertible in

S

.

s = p^+ -«‘+Pn > and to complete the proof of factoriality S , we show that each

a,b e S

are such that

which implies either and

D[S]

is a monomial, necessarily is a unit of

is factorial by

is factorial,

Write

of nonunit prime elements of

D

p^

a + b e p^ +

X*

b e p i + S . Therefore

pi

S .

is prime and

Moreover,

S K[S]

because they are quotient rings of COROLLARY 14.8.

S .

e (XPi ) and aepi + S

Theorem 13.7 implies that cyclic submonoids.

is prime in

Thus, assume

Then or

X aXb

X** e (XPi )

S is factorial.

satisfies and D [G ]

a.c.c.

on

are factorial

D[S] .

Under the notation

e (X^i),

of Theorem 14.7,

179 D[S]

is factorial if and only if Proof.

D and

K[S]

are factorial.

In view of L14.7), we need only show that

is factorial if

D

and

K[S]

are factorial.

Thus,

D[S] S

is a

GCD— monoid by Theorem 14.5, and the same result shows that D[S]

is a

part

(2)

GCD—domain.

Since

of Theorem 14.6,

a.c.c.p.

holds in

it follows that

D [S ]

D[S]

by

is fac­

torial. It turns out that the three necessary conditions and

on

S given in Theorem 14.7 are also sufficient for

toriality of proof.

D[S]

D

and

S

are factorial and let

the group of invertible elements of S

fac-

, but additional work is required for a

Assume that

know that

D

is isomorphic to

ternal weak direct sum of

a

S .

H

be

From Section 6 we

H © M , where copies of

M

is the ex­

, with

a

the

cardinality of a complete set of nonassociate prime elements of

S .

Hence

D[S]

ring over

D

Therefore

D[M]

- (D[M])[H]

is factorial,

should be factorial.

monoids.

D[M]

is a polynomial

in a set of indeterminates of cardinality

H

a .

and we seek conditions on

in order that the group ring of

condition on

, and

H

H

over a factorial domain

According to Theorem 14.7, a necessary

is that it satisfies

a.c.c.

on cyclic sub­

The next two results are concerned with this

condition in an abelian group. THEOREM 14.9. Let

Assume that

G

is a group and

M be the set of positive integers

is solvable in (1)

G

(that is,

The set

M

g

n

g e G .

such that

is divisible by

n

nx = g in

G)

.

is closed under taking positive divisors

and under taking least common multiples.

180 (2)

The following conditions are equivalent. (i)

M

is finite.

(ii)

M

contains a largest element

(which is

is necessarily the least common multiple of the elements of

M) (iii)

M contains only

and for

each

p e M , there is a largest power

p

is in M

that

(3)

If

M

finitely many primes

p

p

of

.

is finite, then

the element

g

has

infinite

order. Proof.

(1):

If

each positive divisor

n e M , it is clear that d

of

n .

each prime-power divisor of show that

lcm(n,m}

case where

n

nx = my = g

and

is in m

and that

and

1 = na + mb .

n,m € M , then

is in

M .

Hence, to

Then

Assume that g = nag + mbg =

.

The implications

(i) ==> (ii) (iii) ~ >

and

(ii) = >

(iii)

(i)

follows from

• To prove

then

for

M , it suffices to consider the

are patent, and the implication

(1 )

m

if

are relatively prime.

namy + mbnx = mn(ay + bx) (2):

n

Thus,

d e M

(3) , we note that if

(g) = (rg)

prime to

k

g

has finite order

for each positive integer

so that each such

r

is in

M

r and

relatively M

is not

finite. If

G

and

g

are as in the statement of (14.9), then

the terminology of abelian group theory is that type

(0,0,...

)

if the set

M

g

is of

of Theorem 14.9 is finite.

To a reader seeing this concept for the first time, the

k ,

181 choice of terminology must seem abstruse, but in fact, the type of an arbitrary element

h e G

can be defined (see

[48, Section 85]); we have no occasion to use the general concept of type. between type

The next result indicates the connection

(0,0,... )

THEOREM 14.10.

and factoriality of

Assume that

G

D[G]

.

is a torsion— free group.

The following conditions are equivalent. (1)

G

satisfies

a.c.c.

on cyclic submonoids.

(2)

G

satisfies

a.c.c.

on cyclic subgroups.

(3)

Each rank— one subgroup of

(4)

Each nonzero element of

Proof.(1) < = = > (2) : follows since tains

G

G G

properly contains

t = ns (t)

is satisfied and let

is of type

The implication

is torsion— free:

^ , then

is cyclic.

.

if

for some

(2) = = > ( 1 )

<s>^

properly con­

n > 1 , and hence

(s)

For the converse, assume that

(1)

(g^) £

- **'

quence of nonzero cyclic subgroups of g l = n lg 2 =

(0,0,... ) .

n i)(~ 8 2) •

be an ascending se­ G .

We have

Hence by replacing

if necessary, we may assume that

n^

g2

is positive.

- g2

Similarly,

after a possible replacement of

g^

that

Without loss of generality,

g 2 = n 2S 3

assume that

(gk+i ) =

n2 > 0 .

g. = n.g. , 6i is i+l

^ 9
with

" ***

such that , and

’ so

for each (1)

by

by

- g^ , we may assume

i , where ’

n. > 0 . l

Then

implies that there exists

n^ = 1

for each

i ^ k .

(2)

is satisfied.

Thus

(g^) =

Corollary 2.8 shows that each rank— one group is the union of an ascending sequence of cyclic subgroups. implies

(3) .

Therefore

(2)

182 (3) = > let

(4) :

Let

H = {x e G|nx e (g)

rank—one subgroup of of

H .

g = m

Since

mh .

in

G

be a nonzero element of for some

since

Replacing h

(g)

If

k

by

-h

Write

H

H = (h)

if

and

is a

is cyclic.

Write

necessary, we assume that

is any positive integer such that

x^ = qh .

is torsion— free, m = kq

x^

Then and

of this equation is

g = mh = kqh , and since k £ m .

Therefore

g

is of

(0 ,0 ,... ) . (4)*— > (2) :

If

(2)

fails, then

strictly ascending sequence subgroups of type

G .

(0,0,...

(g^)

It is clear that

) .

Suppose that

there exists a

< (g2)< ... each g^

of cyclic fails to be

D

is factorial and

G

G

goal is to prove that

is factorial.

D[G]

is a group such

is of type (0,0,...

field, and Theorem 14.15 is that result:

D[G]

is a field and each nonzero element of

(0,0,...

) .

) .

Our

Corollary 14.8

shows that it suffices to consider the case where

D

of

This completes the proof of Theorem 14.10.

that each nonzero element of

if

G

is a maximal free subset

is solvable, then a solution

(h) .

type

n e Z+ }

(3) is satisfied,

is positive.

kx = g

G

g

D

is a

is factorial G

is of type

In preparation for the proof of (14.15), we

first record a result that follows from the proof of part (2)

of Theorem 14.6. THEOREM 14.11.

D

and that

taining ments of

S

T

Assume that

satisfied in

is an extension domain of

is a torsion— free cancellative monoid con­

as a submonoid. S

J

are D

Assume that noninvertible ele­

noninvertible in T .

If

and in J [ T ] , then it is

a.c.c.p. satisfied in

is

D[S].

183 COROLLARY 14.12. field

K

K[G]

and that

H

is factorial, Proof.

and

F[H]

F[H]

Assume that

is a subgroup of the group

then

F[H]

satisfied

If

a.c.c.p.

F[H]

is a

GCD—domain,

by Theorem 14.11.

Therefore

is factorial.

pure in

G

if

H

of an abelian group

G/H

form

nx = h , where

is a subgroup of G/L , then

rated by G

Z+

and

G , and

if

L*/L

is said to be

of an equation of the h € H ,is in

containing

L*

for

is the smallest pure

some

is equivalent

and if

G/H

summand of

If

H

gene­

subgroup ge G

G

is of type (g)*

is

is a pure subgroup of

is finitely generated, then G .

G

n e Z+ . The condition that

to the condition that

g e G .

L

is the torsion subgroup

each nonzero element of a torsion— free group

cyclic for each

If

L , and it consists of all elements

ng e L

(0 ,0 ,... )

H.

is called the pure subgroup of

L . Obviously

such that

Xq

n e

L*

G

is torsion— free; this is equivalent to

the condition that each solution

of

G.

is factorial.

Theorem 14.2 shows that

A subgroup

of

F is a subfield of the

H

G

is a direct

This is the main result about pure subgroups

that we use, and the result itself follows since

G/H

is

torsion— free and finitely generated, hence free. THEOREM torsion— free ments of

p

Let

group

G .

F[H]

Proof. that

14.13.

Let

H

be

If F

a pure subgroup of the is a field, then

are prime in F[G] pbe a prime

divides a product

fg

prime

e le­

.

element of

F[H]

in

sayfg = ph ,

F[G] —

, and assume

184 where by and

h e F[G] .

Let

H u Supp(f) u M/H

M be the

Supp(g) u

Supp(h)

is finitely generated, so

subgroup

of

and hence in

(FtHDfM^] ^ F[M]

F[M]

subgroup of

M .

, it follows

hence in

F[G]

The element

that

.

p

.

. Then

This proves that

H

generated is pure in

M = H © p is Since

divides

G

for some

prime in p

f or g

divides in

p is

M

F[M]

prime in

F[H]

,

fg

in

, and F [G ] •

Most of the work required for the proof of Theorem 14.15 is contained in the proof of the next result. THEOREM 14.14. let

G

Let

F

be a torsion— free group such that each nonzero ele­

ment of

G

is of type

(0,0,... ) , and let

nitely generated subgroup group.

of

G

such that

Then each prime element of

a finite product of prime Proof.

Let

p =

p

F[H]

elements of

l

Since

X*1!


integer by which

F[G]

F[H]

h^ = 0 . hn

Let

p = f1f2 ...ft , then

k

lows that F[G]

p

F[G]

k .

and

<

.

F[G]

is a

let

on

G .

We prove that

F[G] F[G]

such that

Once this is proved,

GCD-domain,

it fol­

elements of

irreducible ele­

are prime (Theorem 6.7).

Thus, assume that there exist nonunits

p

by showing

is a finite product of irreducible

, and since

ments of

.

be the largest positive

are nonunits of t <

a torsion

, we assume without loss of

is divisible in G

f^,f2 ,...,f

a fi­

p , where

is a finite product of prime elements of that if

G/H is

under a fixed to-tal order

n

be

be a prime element ofF[H]

is a unit of

generality that

H

can be expressed as

he the canonical form of

h n < h 0 < ... 1

be an algebraically closed field,

f^,...,ft

of

185 F[G] , with the

t > k , such that

M

subgroup of = Z™ for some

p =

G

generated by

w

, and hence

.

H

and

F[M]

Let

M

u^SuppCf^) is

be Then

factorial,

Con-

sider the following diagram: F [M] -----

F(X;M)

F[H] -----

F (X;H)

In the diagram,

F(X;H)

fields of

and

F(X;M)

F[H]

is

and

F[M]

F(X;M)

, respectively.

finite and normal over F(X;H)

is integrally closed, and that of

F[H]

in

F(X;M)

.

integrally closed. of i

M .

Then

between

that

F[M]

It follows that Xg i

Yn i - Xn igi F

over

F(X;H)

contains the

.

Since

n^th

F(X;M )/F(X;H)

F(X;M)

.

a

6 F[M] T

and hence

F(X;M)[Y]

is finite and normal.

F(X;H)

is a root of unity in

f = group

over

of f

and if

F(X;M) and

over

a(f)

a

pure

is alge­

.

We have shown F[M]

is the integral closure of

Xg

such

roots of unity, and

Moreover, we have also shown that if

the conjugates of where

F

is integral over F[H] , and since F[M]

n^

is a root of the

Therefore

tegrally closed,

is

Moreover, for each

splits into linear factors in

F[M]

F[H]

be a finite set of generators

Yn i _ xn igi

that

F[H]

m , there is a positive integer

€ H .

braically closed,

, that

Theorem 14.2 implies that

Let

and

We prove that

is the integral closure

F(X;M) = F (X;H)({Xgi} .

1

n^g^

equation

denote the quotient

is

F[H]

in­ in

g e M , then

are of the form

F ; in particular,

aXg ,

if

is an element of the Galois

F(X;H)

, then

a(f) = E^ a^ a t X ^ ) ,

have the same support.

186 Since

p

can be factored in

nonunits, where zation of where

p

w

gv . by

in

p^

F[M]

as

is of

the form

F[M]

and where

X g0q^ , where

Cq-^)

is a minimal prime ideal of

n F [H ] = pF[H] F(X;M)/F(X;H) domain,

is normal and

set of prime ideals of (p^)

e^+ e 2 +. .-+er ^ t

of

(p)

0

and degree

F[M]

.

.

generated

Moreover,

by the lying-under theorem, and because

it follows that

Thus, each

t

p = wp®l...p®r ,

has order

It is clear that the ideal

q^

as a product of

t > k , it follows that the prime factori­

is a unit of

We write

F[M]

F[H]

is an integrally closed

(cr((q^)) = (a(q^))|a e T} F[M]

lying over

is of the form

pF[H]

(a^Cq^))

is the

in

F[H]

for some

ck

.

e T .

Therefore

(P)= (pf1 .**p®r) = ([OjCqj)]®1 .••[ar (q1)]er) , so that there is a unit

uXg

of

F[M]

such that

p = uxS[a1 (q1)]e l...[ar Cq1)]e r .

Our previous

observations show that each ^ ( q ^ )

0

gv .

and degree

it follows that

Since

g = 0

and

p

has order

0

has order

and degree

hn = (e^+...+er)gy .

Because

e^+...+er > t > k , this contradicts the choice of Hence

t £ k

as asserted, and

prime elements of THEOREM torsion— free of type

14.15.

k .

is a finite product of

.

Assume that

F

is a field and

group such that each nonzero element of

(0,0,...

Proof.

F[G]

p

) .

hn ,

Then

F[G]

G is a G

is factorial.

In view of Corollary 14.12, we assume without

is

187 loss of generality that nonzero nonunit generated by G

f

Supp(f)

generated by

H .

is factorial and F[H]

.

of

f

F

is algebraically closed.

F[G]

.

Let

be the subgroup of

and let

H*

Since

isfinitely generated,

H

be thepuresubgroup of F[H]

Theorem 14.14 shows that each prime element of

finite product of primes of that prime elements of f

quently,

F[H*]

F[H*]

F[H*]

.

; thus,

f

F[H]

is a

But Theorem 14.13 shows

are also prime in

is a finite product of primes in F[G]

G

is a finite product of prime elements of

is a finite product of primes of

hence

H

Take a

F[G]

F[G]

.

, and

Conse­

is a factorial domain.

The next theorem summarizes the main results of this section.

Most of the work required to prove Theorem 14.16

has already been done. THEOREM 14.16. domain and that Let

H

S

Assume that

D

is a unitary integral

is a torsion— free cancellative monoid.

be the group of invertible elements of

S .

The fol­

lowing conditions are equivalent. (1)

D[S]

(2)

D

is

and

factorial.

S

are factorial

and

S

satisfies

a.c.c.

on cyclic submonoids. (3) of

H

D

is of type Proof.

(2)

and

S

are factorial (0,0,...

) .

Theorem 14.7 shows that

is satisfied,

cyclic submonoids,

then clearly

H

so each nonzero

(0,0,...

)

If

is satisfied,

(3)

and each nonzero element

by Theorem 14.10.

(1)

satisfies element of

Therefore

then since

implies

S

(2)

H

(2) . a.c.c.

If on

is of type

implies

is factorial,

(3) .

D[S]

is

188 isomorphic to Therefore J

.

J[H]

J

, where

J

is factorial.

is a polynomial ring over

Let

Theorem 14.15 shows that

J[H]

K

K[H]

D .

be the quotient field of is factorial, and hence

is factorial by Corollary 14.8.

This completes the

pr oo f. As indicated in the introduction of this section, we do not determine conditions under which But for a group

D[S]

satisfies a.c.c.p.

S , such conditions are given in the next

result. THEOREM 14.17. group ring satisfies type

D[G]

be a torsion— free group.

The

a.c.c.p.

D

if and only if

and each nonzero element of

G

is of

(0 ,0 ,... ) .

isfies

Theorems 12.2,

a.c.c.p.

(0,0,...

that

G

satisfies

a.c.c.p.

Proof.

let

Let

K

)

if

13.7, and 14.8 show that

and each nonzero element of D[G]

satisfies

be the quotient field of K[G]

a.c.c.p.

satisfies

a.c.c.p.

a.c.c.p. D . Since

G

D[G]

sat­

is of type

For the converse,

Theorem 14.15 implies D

also satisfies

, it then follows from Theorem 14.11 that

is satisfied in

D

a.c.c.p.

.

Theorem 14.15 can be used to give examples of finite— d i ­ mensional non-Noetherian factorial domains of any dimension r > 2 .

(A one— dimensional factorial domain is a

hence Noetherian.) decomposable — abelian group of type

For each

r ^ 2 , there exists an in­

hence non— finitely generated — Lr

(0,0,...

PID ,

torsion— free

such that each nonzero element of )

[48, Section 88].

If

F

Lr

is a field,

is

189 then Theorem 14.15 shows that

F[L^]

main is non-Noetherian since and Theorem 17.1 shows that

is factorial.

This do­

is not finitely generated, d imF[Lr ] = r .

Section 14 Remarks Many of the results of Section 14 were originally proved in the paper

[58].

There is some discussion at the end of

Section 7 of [58] of the problem of determining conditions under which

a.c.c.p.

some aspect of

is satisfied in

a.c.c.p.

D[S]

.

One trouble­

is that it doesn't behave well with

respect to localization in either direction.

For example, if

D = Q[Q]

, then

marks at

the end of Section 13) that does not satisfy

a.c.c.p.

, but Dp

of

On the other hand,

D .

consisting of

D is an almost Dedekind domain (see the

is Noetherian for each maximal ideal if

S

elementary degree argument shows that a.c.c.p. zation of

But F[S]

S

has quotient group , and

F[Q]

P

is the additive semigroup

and the rational numbers

0

re­

q > 1 , then an

F[S] Q ,

satisfies F[Q]

does not satisfy

is a locali­ a.c.c.p.

Another condition frequently considered in connection with

a.c.c.p.

that

D

and factorization properties is the condition

should be atomic, which is defined to mean that each

nonzero element of elements D

(atoms) of

is atomic.

D

is a finite product of irreducible D .

If

D

satisfies

a.c.c.p.

, then

The converse fails, but examples are hard to

come by (see [64] and [135]). conditions under which Section 8 of [97]).

The question of determining

D[S] is atomic is open (see, however,

190 §15.

Monoid Domains as Krull Domains

We continue in Section 15 to denote by tegral domain, and by

S

D

a unitary in­

a torsion— free cancellative monoid.

Krull domains are frequently treated from two points of view; one approach is in terms of valuation overrings, and the other is in terms of divisorial ideals and the

v— operation.

Our treatment follows this separation of approaches.

In this

section we regard a Krull domain as the intersection of a family of rank-one discrete valuation overrings of finite character;

in Section 16, where we determine the divisor

class group of a monoid domain that is a Krull domain, the topics of divisorial ideals and the

v— operation naturally

arise. Throughout this section we generally use various modifi­ cations of the letter while forms of

v

w

to denote valuations on groups,

are used for valuations on fields.

(There is an exception to this practice in the statement of Theorem 15.7, where If

w:G -- »- r

w*

is used for a valuation on a field.)

is a valuation from the group

ordered group

r , then

w(G)

called the value group of if

w(G)

rank

w(G)

family

w .

We say that w

, as an ordered group.

of rank one if and only if ^w a ^aeA

w(G)

valuations on

character if, for each

into the

is an ordered subgroup of

is discrete, and the rank of

of

G

w

is discrete

is defined as the

Thus,

w

is discrete

is isomorphic to G

r

Z .

A

is said to be of finite

g e G , the set

{a e A | w a (g) * 0}

is finite. Assume that that

S

G

is the quotient group of

S .

is a Krull monoid if there exist a family

We say ^w a ^a€A

191 of rank-one discrete valuations on has finite character and

S

G

such that

^wa ^aeA

= (g eG | w a (g) - 0

for each

a e A) ; the latter condition is equivalent to the assertion that

S

is the intersection of the valuation monoids of the

valuations

wa .

In Theorem 15.5 we show that

Krull domain if

D

is a Krull domain,

and each element of of

G

is of type

S

D[S]

is a

is a Krull monoid,

(0,0,...

)

(15.5) uses several preliminary results,

.The proof

but a proof

of

the converse of (15.5) can be given at once. THEROEM 15.1. ments of

S .

If

Krull domain, of

H

Proof. K n D[S]

of

H

be the group of invertible ele­

isa Krull domain,

then

D

is a

is a Krull monoid, and each nonzero element

type If

(0,0,...

K

).

is the quotient field of

, and hence

domain satisfied a.c.c.

H

D[S]

S

is of

Let

D

D , then

is a Krull domain.

a.c.c.p.

D =

Since a Krull

, it follows that

S

satisfies

on cyclic submonoids, and hence each nonzero element is of

group of

type

(0,0,...

S , and let

).

^va ^aeA

crete valuations defining w a *.G -- ► Z

B = {a e A | wfl

has rank one}

f°r each

<— >va (Xg )

a 0

Therefore

a

by .

S

»anc* let

Moreover,

for

ge G ,

for each

a e A <— > X8e D[S]

is a Krull monoid.

For

It is clear that the family

a e B < = > w a (g) > 0

for each

quotient rank-one dis­

w a (g) =

has finite character.

w a Cs) - 0

Gbe the

D[S] as a Krull domain.

a c A , define

{wa )aeB

Let

ae

<=~>g e S

A .

192 If S

H

is the group of invertible elements of

is factorial if and only if

for some cardinal

y .

notation.

with free basis IIj :F -- ► Z

Let

defined by

•

valuation on

The family

ments of

S .

(1)

S

(2) M

nF+

G

mapping

is called the

F+ =

is °f finite char­

(En^e^ | n^ > 0

F Let

H

be the group of invertible e le ­

The following conditions are equivalent.

H © T , where

S

F

is of the form F

T

is

and some subgroup

H © T , with

is a free group and

G

M

of

is

F

.

form

thequotient

T . (1) = > ^a^aeA

and let U

A , and let

ker(f) © G ^

S

f :G -- >- F

, where

(2) :Let

he a family of rank-one discrete valua­

defining

f(g) = Ewa (g)ea .

G^ .

ofthe form

Tof the

as a Krull monoid.

Let

be the free abelian group with free basis

to

,

under the cardinal order.

is of the form

nF+ ,where

tions on

by

i e 1}

for each

Proof. S

j th

the Krull monoid determined

forsome free group

group of

the

is a Krull monoid.

S

(3)

eI »

It is, of course, a rank-one discrete

acter, and we denote byF+

THEOREM 15.2.

3

ITj (En^e^) = nj

F .

the positive cone of

( Z q )

The statement of (15.2) uses

F°r

projection map on

this family; thus

y

H ©

F = ^ ^ Z e ^ bea free abelian group

^e i^i€i

F .

•

is of the form

Theorem 15.2 provides an analogue of

this result for Krull monoids. some new

S

S , then

Since

indexed by

be the homomorphism defined by F

is free, the group

G^ - f(G)

We show that

F = EZea

G

splits as

under the restriction of

H = ker(f)

.

By definition,

f

U

b

193 ker(f) = (g c G | w a (g) = ker(f)

is

for each a e A} .

a subgroup of H

a e A , w a (h) > 0 and

0

h e ker(f)

and

.

.

If

h e H , then for each

w a (- h) = - w a (h) > 0 , so

Therefore

G = H © G^

and

On the other hand,

G^ n Sis isomorphic under

f(Gx) n f CS) = f(G)

n f(S)= f(S) = f(G)

M = f(G)

, this establishes

The implication (3)

implies

(e i )i£i .

For each

defined by {yi>i£I of

(3)

f

to

Taking

and

G

To prove that

as

subgroups of

is free abelian with free basis

i , let

y^.H ©F

has finite character. and

S = H © (G^ n S ) .

n F+ .

isclear.

S

y^(h + Enjej) = n^ .

yA toG

for

(2) = >

F = EZe^

wa (h) = 0

(2) .

(1) » we regard

H ©F , where

Therefore

let

-► Z be the mapping

We note that the family

Let

w^

J = {i e I |

be the restriction has

rank one} .

Then

g= h + Enjej e G , we have w^(g)

^

0for each

<=>

S

each

(3)

S = H © T , where

of Theorem 15.2.

main and that each nonzero )

is a

i

e I

+T = S . G

that d e ­

as a Krull monoid.

Assume that

D[S]

0 for

a family of valuations on

termines

(0,0,...

eH

g - h e G ^ F+ = T<“ > g

Therefore

part

i e J <=>w ^( g) ^

.

Then

T = G n F+

Assume that element of

D [S ]= D[H][T]

D

is as in

is a Krull d o ­

H is of

,and hence to prove that

Krull domain, it suffices to prove that

a Krull domain and that the monoid ring of domain is again a Krull domain.

type

T

D[H]

is

over a Krull

The next result, which con­

cerns extension of valuations on fields, will take care of

194 the case of

D[H]

, and that result can be used to dispose of

the general case. If

V

is a valuation overring of

maximal ideal of center V

of

V

V , recall that

on

D , and that

is a quotient ring of

D

and if

P = M n D V

M

is the

is called the

is essential for

D , in which case

V

D

if

is necessarily

Dp . THEOREM 15.3. of

D

Assume that

with center

quotient

field

P

K

of

the value group of v*: D[S]\ {0} -- >■ r

on

v

canonical form of the

(1) of

v*

D[S]

ring on

(2)

D

of

be a valuation on

associated with

V

.Let

the

r

be

nonzero element f

of

D[S]

, then

.

v*

v*

is

is an overring of

L

r . The valuation D[S]

with center

P [S ]

. v

If

such that

is essential for D[S]

K[S] n (nv^*)

Moreover,

Proof.

if and only if

v*

is

(V } . is a family of valuation overrings of a aeA D = n v , and if v„ is a valuation on K aeA a a Va

{va*}

D

.

associated with

then

v

determines a valuation on the quotient field

essential for (3)

D

; the value group of

V* D[S]

D .Let

is a valuation overring

and define the function n s. follows. If f = E ^ X 1 is the

as

v*(f) = in ff vO i) )!

V

.

for each if

a e A ,

then

D [S ] =

has finite character,

also has finite character. (1):

Let

^

with the group operation.

be a total order on G compatible n s• m t. Let f = E-^a^X and g = E-^b^X 1

be the canonical forms of nonzero elements

f,g e D[S]

, where

195

S1 < s2 < * * * < sn

and

minimal

so that

v*(f) = v(a^)

f + g *

0

and if

f + g ,then

*1 < 12

aj + bj

< • • • < tin * and

Choose

k and

is a nonzero coefficient of

v(aj + bj ) a inffvfa^) ,v(bj)} a inf{v(ak ),v(bu )

inf{v*ff),v*fg)} .

Therefore

v*(fg) ^ v*(f) + v* (g) is clear. Si+t the coefficient of X K u in fg has the form a,b + (a sum of terms k u v x < k

or

proved that clear that D [S ] c V*

Therefore

v*

is the value

holdssince

(2):

v*

a b , where either x y *

v*(fg) = v*(f) + v*(g) .

determines a valuation on

very definitions center of

v

of P

on

D[S]

is essential for

D if

prove the converse.

group of

v* .

v*

The inclusion

imply that

D , and P[S]

Since

D p [S ]

a

is a generator of the ideal

f

can be written as

a/b e D .

is a quotient ring

V*

is essential for

and

V*

= D[S]

(a1 ,...,an )D

af* , where

, where

v(a)

of

v*(f) = v(a)

Thus, an arbitrary

Consequently,

is the

D[S]Drc, n K = D implies that v P[S J P v* is essential for D[S] . We

Thus, we assume without loss of generality that n s\ f = Z^a^X be an arbitrary nonzero element of

is of the form af*/bg*

the

.

D[ S] , it suffices to show that

.

We have

L , and it is

is nonnegative on

and

The equality

f* e D[S]\P[S]

Moreover,

this coefficient is v Ca^) + v(bu ) =

.

r

.

y < u) .

v— value of

v*(f) + v*(g)

} =

v*(f + g) a inf{v*(f),v*(g)}

The inequality

The

u

v*(g) = v(bu ) .If

D[S]

.

.

Let If

D , then and of V*

, and hence

af*/bg* = (a/b)f*/g* e D[S]

r , , as asserted. P[S]

D p [S]

D = V .

nonzero element > v(b)

of

P [S J

,

196 (3): V a [S] .

For a fixed a

O CnaeAV*) = naeACVa [S]) = (nVa ) [S] = D[S]

Assume that

{va }

has finite

be the canonical form

1

some

between

1

* 0}

is

and

n}

f e D[S]

v*(f) = 0 ; hence

The

i between *0

for

a e A\B ,

For

has finite character.

Z^a^XS i .

finitefor each

is also finite.

.

f=

B ■= uV ,B. = (a e A I v (a.) i=l i 1 a l

n , and hence i

character and let

of a nonzero element

B. = {a e A I v (a.) l a i and

n K[S] =

Thus,

K[S]

set

, it is clear that

This completes

the proof. THEOREM 15.4.

Assume that

D is a Krull domain and

is a group such that each nonzero element of (0,0,... ) . Proof.

Then Let

K = D , then K * D , let ations on

D[H] K

defining

D

D[H]

of Theorem 15.3.

then part

(3)

as a Krull domain. and for each

K[H]

If

V*

Moreover,

is a Krull domain, THEOREM 15.5.

ments of

S .

L

Let

Assume that

(0,0,...

) .

Then

L

be the

v*

be

determined as in part

D[H]

Cv a^aeA

= K[H] n (nV*)

v*

,

^as

, andsince

is also a Krull domain.

be the group of invertible ele­ D

is a Krull domain, that

a Krull monoid, and that each nonzero element of type

If

is the valuation ring of

D[H] H

Let

a e A , let

of Theorem 15.3 shows that

finite character.

If

a family of rank-one discrete valu­

the rank-one discrete valuation on (1)

D .

is factorial, hence a Krull domain.

{va }aeA

quotient field of

is of type

is a Krull domain.

be the quotient field of

D[H]

K

H

H

D[S]

is a Krull domain.

H

S

is of

is

197

Proof.

We have already observed that it suffices to

consider the case where G n F + , where group of of type

S .

F

.

Then

is a free group and

Since

S

G

is the quotient

Therefore

Moreover,

D[F+ ]

D[G]

n D[F+ ]

G

is

is a Krull domain by

is a Krull domain since it

is isomorphic to a polynomial ring over D[S] = D[G]

is of the form

G £ F , each nonzero element of

(0,0,... ) .

Theorem 15.4.

H = (0)

D .

It follows that

is also a Krull domain.

Combining Theorems 15.1 and 15.5, we have proved the following summary result. THEOREM 15.6. main and that H

S

Assume that

D

is a unitary integral d o ­

is a torsion— free cancellative monoid.

be the group of invertible elements of

S .

Let

The following

conditions are equivalent. (1)

D[S]

(2)

D

is a Krull domain.

is a Krull domain,

each nonzero element of If

D

H

S

is a Krull monoid, and

is of type

is a Krull domian,

(0,0,... ) .

it is known that the set of

nontrivial essential valuation overrings of where Dd

^pa^aeA

t*ie set

is rank-one discrete,

character, and

D = n

in each family

F

D

is

minimal primes of the family

{D d }

(Dp )a€^ >

D .

Each

is of finite

Dd . The family {Dp } is called A *a *a the defining family for the Krull domain D ; it is contained

of

D

such that

of rank-one discrete valuation overrrings F

has finite character and

D = nF , and

hence is the unique family of essential valuation overrings of

D

satisfying the three properties listed.

Theorem 15.3

and the proof of Theorem 15.4 determine the defining family

198 for

D[H]

follows.

, for D If

and

H

as in the statement of (15.4),

K is the quotient field of

factorial and the defining family for where

D , then

K[H]

K [H ]

complete set of nonassociate prime ele­ K[H]

.Each of these valuation rings

.K [ H ] ^

essential

for

D[H] since

ring of

. Theorem

15.3

K[H]

is a quotient

and the proof of (15.4) show

D [H] = K [H] where

is

is {K[H]

ments of

D[H]

as

n (naeAD[H]p m a

^

is

that

) ,

{P

} is the family of minimal primes of a aeA equality implies that the defining family for D[H]

D .

This

is

{KW ( qi) }i £I U W Hh a[H]K*A ■ For a general Krull domain H © (G n F+)

as in part

D[S]

, where

S =

(3) *of Theorem 15.2, the proof of

Theorem 15.5 shows that the defining family for sists of the defining family for

D[H][G]

those members of the defining family for essential for where

H © G

D[S]

Let

the fact that D[H][F ]

We note that

, together with D[H][F ]

D[H][G]

that are

- D[H © G] ,

is isomorphic to the quotient group of

defining family for paragraph.

.

D[S] con­

L

D[H][G]

is determined in the preceding

be the quotient field of

L[F+ ]

S ; the

D[H]

.

Using

is factorial, the defining family for

can be determined from part

u

(3) of Theorem 15.3 as

[F+ ]Q e[ F+ ] >6eB ,

where

{h. }. T is a complete set of nonassociate prime eleJ J ments of L[F+ ] and where is the set of minimal prime ideals of

D[H]

.

We pursue this line of reasoning no

199 further, for there is a better approach to determining the essential valuations for family for

D[H][G]

.

D[S]

that are not in the defining

The approach alluded to involves prov­

ing an analogue of Theorem 15.3 for valuations on a group. THEOREM 15.7. let

w

Let

G

be a valuation on

monoid

W

contains

f = E^a^X^

of ring

s e S

w*(f) = i n f f y fs ^} ^

is an

, where

D[S]

overring of

T

w*

w(s)

of has

S

Assume that

.

D[S] = D[G]

character,

Proof.

(1) :

of

f

Choose

f

Let

and k

= Z^a^X^ g , where ® and

w*(g) = w (tu ) •

u

with center

D[S\T]

consisting

L

on of all

D[S]

if and only if

£

W

is

T .

a

valuations on

W a is the valuation

n (nW*) .

Moreover, if

monoid {wa >

also has finite character.

be a total

order on

Take nonzero elements

G

compatible f,g € D[S]

,

andg = E^b^X**-be the canonical forms s n < s 0< . . . < s 1 2 n

and

t,< . . . < t 1 m

minimal such that w*(f) = w(s^) If

f

r . The valuation

then{w* }

with the group operation. let

If

quotient field

with respect to

, where

finite

and

is

^w a ^a€A

S = naW a

Then

w* D

is essential for

such that w^

as follows.

w

= 0 .

the quotient monoid of (3)

of

.

is the submonoid ofS

such that (2)

G

be the value group

w*:D[S]\{0} -- *■ r

; the value group of

W*

and

whose associated valuation

w*determines a valuation on the

D[S]

S

is the canonical form ofthe nonzero element

S , then (1)

G Letr

S.

and define the function

of

be the quotient group of

f * - g

and if

s € Supp(f + g) c Supp(f) u Supp(g)

, then either

and

200

w(s)

>

or

w fs) >w(tu ) .

w(s)

> inf {w*(f),w*(g)}

, and hence

w*(f + g) > inf {w* (f) ,w* (g)} w*(fg)

>w*(f)

holds since efficient of Therefore T . on

+ w*(g)

s^ + t

.

The inequality

is clear, and the reverse inequality

e Supp(fg)

X sk+tu

w*

Either way,

in

fg

; to wit is t^e co­

by choice of

determines a valuation on

The inclusion

D [S ] c W*

and u

Assume that

g € W , write There exists

w*

X g = f/h

w

is nonnegative

it follows that

Thus

is essential for

D[S]

g + t € S , and hence Y

system in that

W*

D[S]

of

S

consisting of units of

it is essential for D [W ]

w(a),

W*

and where D[W]

D[S]y = D[W]

.

w*(f*) = 0 .

is of the form f*

.

, and

and

g*

W* = D[Wlw Mq (3) : For a fixed

=

t be­ T

.

For the

is a multiplicative W* .

Thus, to show

X a f* , where

f

w*(f) =

Thus, an arbitrary ele­ , where

w(a) > w(b)

M Q , the center of

W*

X af*/Xb g* = X a 'b f*/g* « D[W]

"o

and hence

a > we have

D[G]

n W* =

Therefore D[G]

f ,

Each nonzero element

X af*/X^g*

are not in

It follows that

X gh

.

, it suffices to show that

is expressible in the form

f* € D[W]

ment of

D[S]

For

with respect to

U = {X1* | t € T}

is essential for

.

g = (g + t) —

W c y , and the reverse inclusion is clear.

converse, we note that

is

f e D[S] , h $ D[S\T]

n T . From the equality

longs to the quotient monoid

w*

w* .

for some

t e Supp(h)

.

with value group

S , and the assertion concerning the center of

(2) :

on

L

holds since

immediate from the definition of

of

k

n (naW*) = naD[Wa] = D[nWa] = D[S]

.

,

201

Assume that Z?aiXSl

{w a )

has finite character and let

be the canonical form of a nonzero element

f e

D[S]

the

set B= (a e A |wa (si)

n)

is finite, and

{w*} a

f =

.Asin the proof of part * 0

w*(f) = 0

(3)

of

forsome

i

between

a

for each

Theorem 15.3,

e

1

and

A\B .

Thus

if

is a

is also of finite character, If G

is the quotient group of

valuation on then we say

G

respect to

and

whose valuation monoid

that

essential for

S

W

w is essential for

S)

if W

is the

S

w

contains

S ,

(and that

W

is

S

with

quotient monoid of

T = {s e S | w(s) = 0} .

The next result

follows

from Theorem 15.7. THEOREM 15.8. quotient group of type

G

Assume that

S

is a Krull monoid with

and that each nonzero element of

(0,0,... ) .

^ct^aeA

Let

S

character.

.

Then S = ^ a e p ^ a > and

If

^U3 ^ 3 €B

valuation monoids on n$eBus * then

< V

£

G

{V

that are essen­ {Wa >

*s any fami!y

G

is

°^

rank-one discrete valuation monoids on tial for

G

has finite

°f rank-one discrete

of finite character such that

S =

•

Proof. Let w be a valuation associated with W , u ----a a 3 a valuation associated with ,and let K be a field. If F

is the defining family forK[G]

Theorem 15.7 shows that for the Krull

domain

is

r

F u {U*}

S) .

i

Y

We have

, then

F u K[S] . The

, where

part

of

is a defining family defining family for

C = {3 e B | U*

p

is essential

K[S] = K[G] n (ny£CU*) = K[ny UY ]

proof of Theorem 15.7), so that

(3)

S =

•

K[S] for

(see the

202

)E

*s clea r*

On the other hand, each

W*

essential rank-one discrete valuation overring of hence belongs to the defining family of

W*

on

D [S]

is

D[S\Ta] , where

This ideal extends to K[G]

K[G]

for each

s e S \Ta

W* = U*

for some

hence

k[ G] n U* = K[U ] , so L

y

F u {U*} .

.

a

, for

Xs

It follows that

W*

= U

Thus

.

D[S]

, and

The center

T a = {s € S |w a(s) * 0} .

K[G]

e C .

Y

W

Y

in

is an

K[G]

is a unit of { F , and W* = K[Wa] =

n

We conclude that

{U } = Y

y

(W^} , and this completes the proof. COROLLARY 15.9.

Assume that

The defining family for

{ K rG V

)

D[S]

D[S]

is

}i £I U {D[S]P [ S ] }a eA u {W6 }6£B > a

where the notation is as follows: of

D ,

G

is a Krull domain.

K

is the quotient field

is the quotient group of

S ,

is a com­

plete set of nonassociate prime elements of

K[G] , and

{Pa )aeA

is the set of minimal primes of

{Wp}peB

is the family of rank-one discrete valuation monoids

on

G

that are essential for

D ; finally,

S .

Several remarks on the statement of Theorem 15.8 and Corollary 15.9 seem appropriate. valuation monoids on following reasons. on

G

Note that ments of

W

of

ker(w) W .

Thus

(15.8) is stated for

G , rather than valuations, If

w

with value group

tion monoid

First,

w

is a rank-one discrete valuation dZ , where

is

for the

d > 0 , then the valua­

ker(w) © ^ » where

w(g) = d .

is precisely the group of invertible ele­ W

does not determine

rather up to the value of

w

on

g .

w

uniquely, but

This wee difficulty

203 can be avoided by assuming that all rank-one discrete valua­ tions dealt with have value group said to be normed) .

Z

(such a valuation is

Working with normed valuations, Theorem

15.8 can be rephrased by replacing and

(up)

, respectively.

neither of the families

D [S ] p j-gj

(K[G] ^

^ ^ ei

’ a description in terms of

tained:

K[ G] (q.} = D[S ](q.)nD[s]

(S\Tg}geB If of

S

{wa )

.

is described

D[G]p

.

With

D[S]is easily Similarly,

ob-

W* =

Theorem 16.10 of Section 16 we show that

is the set of minimal prime ideals of S

by

; in the other d i ­

could be written as

] > aRd

(Ug)

or

D[S]

K[G](q.)

D[S] d [s \t

and

In the statement of Corollary 15.9 ,

implicitly in terms of the domain rection,

{Wa )

is finitely generated,

is also finitely generated.

S .

then the quotient group

G

The converse need not

hold in general, but we show in Theorem 15.11 that the con­ verse holds if

S

is a Krull monoid.

An ancillary result

is used in the proof of (15.11).

THEOREM 15.10. Let A be an infinite subset of v where F = Zl^Ze^ is a free group of finite rank k . by

<

the cardinal order on

F .

Then

A

finite strictly ascending sequence under Proof. is patent.

We use induction on In the case where

k

k .

that

Denote

contains an in­ £ .

The case

where

k = 1

> 1 , let M =

n^(A)

be the

set of first coordinates of elements of infinite for some

F+ ,

A .

If

Il^fm)

is

m e M , then the induction hypothesis implies

JI^1 (m) , and hence

* increasing sequence.

A , contains an infinite strictly

On the other hand,

if

-1

(m)

is finite

204

for each A

m € M , then there exists an infinite subset

such that distinct elements of

n i ^ 2^ < ' ’ ’ *

If

Let

B

ci = ^i ~~

f°r each

is a finite set, then there exists

such that the set {bi>iei

of

have distinct first oo Thus, we may assume that B = { b ^ ^ , where

coordinates. ni^l^

B

I = {i e Z+ | c. = c}

i •

c e {c^}”

is infinite, and

is aninfinite strictly ascending sequence in

In the contrary case, where

A .

is infinite, the induction

hypothesis yields an infinite strictly ascending sequence c^

< c^

< ...

of the set

.

Let

{II^ (b

^

the smallest integer Then

= i^ .

b. < b. J1 J2

Only finitely many elements

are less than i^

) .

such that

and we can

ni ^ i t ^

We let

jT j

II-. (b. ) > n, (b. ) . 1 ^-u 1 J2

{

be the smallest

Then

j2

OO Cb ^ a r e i u

be

(b . ) .

>

repeat the argument on

wit, only finitely many elements of II^ Cb. ) . 1 j2

Let

b. .To J2 less than

such that

b. < b. < b. , and a continuaJ1 ^2 J3

tion of the process yields an infinite strictly ascending sequence

(b. } Jr

in

A .

The result follows by the principle

of mathematical induction. THEOREM 15.11.Assume quotient group then

S

S

is a Krull monoid

with

G is finitely generated as a group,

is finitely generated as a monoid.

Proof. family

G .If

that

Assume that

G

is generated by

{wa^a€A °f nontrivial valuations on

character is necessarily finite since finite for each

i

defining family for

between S

1

and

is finite.

{g^ G

.

A

of finite

{a £ Alw^fg^) * 0 } n .

is

In particular, a

The proof of Theorem 15.2

205 shows that

S

is of the form

of invertible elements of where

F

H © T , where

S

and T

F . Since

H

s G, H

as a group and as a monoid.

is of the

shows that the set Let

U

B

be the submonoid

less than

t , so

t e U .

Consider

If

b^ . If

other case, T .

F

of

T c U.

Thus, take

t * b^

, then

t > t -b^

t - b^

t > t — b^

and

. To

a

non­

F+

M n F+ = such that

= b2 , w e ’re finished, and

in the

> t - b ^ — b 2 is a decreasing

sequence

Again because only finitely many elements of

Therefore

T = U , and this completes the proof. T

are

t = b^ ,

t

b2 e B

is

B

for some element b^cB . I f

t , it follows that

free on

T

Only finitely many elements of

t > b^

T

Theorem 15.10

T generated by

less than

If

is a sub­

as a partial­

< .

Repeating the process, there exists

b2 < t -

in

t e T .

M

of nonzero minimal elements of

complete the proof, we show that zero element

M n F+ ,

Hence, it suffices to show that

ly ordered group under the cardinal order

T .

form

group

is finitely generated both

is finitely generated as a monoid.

then

is the

is a finitely generated free group and

group of

finite.

H

is of the form

M

t = b-^ +

...+b^ € u

n F+ , where

{ea ^aeA » then the monoid

domain

garded as a subring of the polynomial ring

F+ are forsome

F = ^ 0LepiZea D[T]

is

can be

D[{X }

*]

Ot Ot € A

k.

re ­ over

D .

Moreover, D[T] is generated as a ring over D by "pure 0 6 e monomials” X *X 2 ...X n , with e^ £ 0 for each i . Conal a2 an versely, each ring

, where each

monomial in the indeterminates

Xa , is of the form

where

U

is a submonoid of

m^

is a pure

F + . Rings of the form

t>[U] , D[{m^}]

arise as objects of natural interest in various parts of commutative algebra.

Thus,

it seems worthwhile to state the

206 following consequence of Theorem 15.11 as COROLLARY 15.12.

Assume

that

D is

grally closed domain and that indeterminates over

D .

Let

{ma ) » and let

separate result.

a Noetherian inte­ a

{m a }aeA

monomials in the indeterminates generated by

a

finite set of 3 S6t

X. , let l *

T

J = D[{ma >] .

Pure

be the monoid The following

conditions are equivalent. (1)

T

is finitely generated and integrally closed.

(2)

J

is Noetherian and integrally closed.

(3)

J

is a Krull domain.

We remark that under the notation of Corollary 15.12, integral closure of

T

is finitely generated. D +XD[X,Y] each

does not, in general, For example,

imply that

D[{XY1}~= q ] =

is a non-Noetherian integrally closed

integrally closed domain

T

domain for

D .

Section 15 Remarks Three general references on Krull domains are [20, Ch. VJI], [44], and [51].

A number of other classes of domains that

are related to Krull domains have been considered in the literature.

Among classes whose definitions involve certain

conditions on the family of valuation overrings of the given domain, there are the classes of domains of finite character, domains of finite real character, domains of finite rational character, domains of Krull type, and generalized Krull do­ mains

(for the definitions, see Section 43 of [51]).

Each of

these concepts has a natural extension to monoids, so that one could consider the classes of (always torsion— free and

207

cancellative) monoids of finite character, etc.

Using

Theorems 15.3 and 15.7 and other techniques from this section, it can be shown that the monoid domain named classes if and only if S

D[G]

D[S]

is in one of the

belongs to the class, and

belongs to the monoid class of the same name; here

notes the quotient group of

S .

0 , Matsuda in [97] has shown that if and only if of type

D

(0,0,...

For

D

D[G]

G

of characteristic is in a given class

is in the class and each element of ) .

For

D

de­

of characteristic

G

is

p > 0 , the

conditions are almost the same; the difference in this case is that divisibility of nonzero elements of powers of

p

G

by arbitrary

does not affect whether or not

D[G]

belongs to

the class (see [100] for details). Another class of domains related to Krull domains are the

n—domains, where

D

each principal ideal of ideals of

D .

II— domain iff invertible iff D

D

is defined to be a D

is a finite product of prime

According to Theorem 46.7 of [51],

is a

is factorial for each maximal ideal

Anderson and Anderson

D

D

M

of

are finitely generated.

[5] have shown that

n— domain if and only if

is a

D[S]

II—domain,

and each element of the quotient group of ) .

D

is a Krull domain whose minimal primes are

and minimal primes of

(0,0,...

n— domain if

S S

is a is factorial,

is of type

208 §16.

The Divisor Class Group of a Krull Monoid Domain

Theorem 15.6 gives necessary and sufficient conditions for a monoid domain

D[S]

to be a Krull domain, and subse­

quent results in Section 15, such as Corollary 15.9, determine the defining family for primes of

D[S]

.

D[S]

and the family of minimal

Using heavily these results from Section

15, we determine in Corollaries 16.7 and 16.9 the divisor class group of

D[S]

.

To begin, we review some of the basic

terminology, results, and notation concerning the

v-operation

on an integral domain, and in particular on a Krull domain. Later in the section we replicate much of this development for cancellative monoids, assuming no familiarity on the part of the reader with the results in the context of monoids. Let K of

D

be a unitary integral domain with quotient field

and denote by D .

For

F(D)

the set of nonzero fractional ideals

F e FCD)

,F~*

is defined to be

{x e K | xF c D} , and the fractional ideal noted by

Fv .

Equivalently,

Fy

F -- ► Fv

is called the

and a fractional ideal v— ideal)

if

A v = Bv .

div(A)

noted by

that contain

v— operation

of

on

v—operation on on

F(D)

V (D)

, and

.

D ,

(or a

defined by setting A ~

D , the class of

Under the operation

~

are called

A e F(D)

is denoted D

div(A) + div(B) =

is a group if and only if

D

is

B

is de­

is a commutative monoid with zero element P(D)

F .

Dinduces an

, and the set of all divisor classes of

div(AB) , V (D) div(D)

D

The equivalence classes under

divisor classes by

~

is de­

is said to be divisorial

F * Fy . The

equivalence relation if

F

(F *) *

is the intersection of the

family of principal fractional ideals of The mapping

D:F =

209 completely integrally closed. {div(xD) | x e K,x * 0} ible elements of

P(D)

F e F(D) of

C(D)

, and C(D) = D ; if

D

P(D)/P(D)

[div(F)]

determined by

div(F)

of invert­

is

called the

is completely integrally

is the divisor class group of

, we denote by

C(D)

P(D) =

is a subgroup of the group

divisor class monoid of closed, then

The set

the element

D .

For

div(F) + P(D)

.

We now specialize the discussion of the preceding para­ graph to the case of a

Krull domain.

a Krull domain, that ideals of

Thus, assume that

D

is

the family of minimal prime

D , and that

v^

is a normed valuation on

K

associated with the valuation domain i e I .

V. = D_ for each 1 pi In this case it is known that fciivCP^D JicI

free basis for the group Fy =

> and

scripts

i .

FV^ =

Moreover,

%

P(D) .

In fact, if

a

F e F(D)

, then

forall but a finite set of

for an integral ideal

A

sub­

we have

■

is a finite intersection of symbolic powers of the minimal primes

P^

of

D

that

contain A .

The first

section relates the divisor class group of quotient overring THEOREM 16.1. Krull domain F(D^)

D .

induces

The kernel of prime of

D

Proof.

of Let

N

be a multiplicative

The mapping

F -- ► FDN

is generated by

that meets

to that

of

a

D .

a homomorphism <J>

D

result of the

of

C(D)

of

system in the F(D)

onto

into

C(Djyj) •

{[div(P)] | P is a minimal

N} .

Consider the mapping

a:P(D) -- »■ P(DN )

defined

210

by

a(div(F)) = div(FDN )

for

F e F(D)

is well-defined, we first show that exists an integral ideal of

D

such that

d 1 (ADn )v . Ay D^ .

A

of

F = d 1A .

Hence

meets

\

=

j .

contain

v

= d *A

v

containing

A

meets

of minimal primes of

nT=1pjn P

D

containing

{pjDN^i

, where

ADp

= PjjDp

=

= nJ c P j V (nj) •

the set of minimal primes of

P;iDp.= CPjDp,)n j CDN )p.n

j

div(F) = div(G)

J

j

1

, then

j

j N

= AyDN .

Fy = Gy ,and hence

is well-defined.

It is clear that

Thus,

if

div(FD^) =

a

This shows

preserves

is surjective since each integral ideal of D .

Since

PCD)

free on (divCPD^)

follows that o

g

is extended from

and

that

for each j .

div((FDN )v ) = divCFyDN ) = div(GvDN ) = divCGDN ) .

Dn

DN

and

Therefore (ADN )y = nj=1 CPj D„) (n 35

addition, and

A

We have

j N

a

(divCP)| P

meets

|P N}

is free on does not meet

(div(P)} N} , it

generates the kernel of

. Since

,

P. does not meet N for 1 < j £ k , J J N for k + 1 < j < r . Then

(ADN )CDN)p.D = ADP.=

that

N

(ADXT) = DXI = A DXI . Thus, assume v N'v N v N *

nI (pjnj)V

ADXT N

D

F

d

(ADN )v =

is labeled so that

Moreover,

and a nonzero element

Thus, it suffices to prove the equality

{P^

a v dn

There

(FDK1) = N v

that the set

for each

*

a

and

then it is clear that

Pj

To show that

(FD^)v =

D

If each minimal prime of

while

.

a(PCD)) c PCDn ) , then

a

induces a surjective

211

homomorphism

<J>: C(D) -- »- C(D^)

[divfFD^)]

.

Suppose

Then

= yD^

hence

[div(F)]

so

divfy *F)

divfy ^F) = d i v ( P ^ .. .pjjm )

{ki>i .

D

meeting

It follows that

that meets

N}

e ker a

for some set

N

<j> .

, and

{P^

of

and some set of integers

[div(F)] = [div(P^1 ...P^m ) ] =

Z^ki[div(Pi)] , and hence D


belongs to the kernel of

is principal,

minimal primes of

of

defined by

{[div(P)] | P

generates

is a minimal prime

ker <J> , as asserted.

There are strong similarities between the proofs of Theorem 16.1

and the next result, which relates the class

group of a Krull

domain

THEOREM 16.2.

D[S]

to that of

D .

Assume that the monoid domain

D[S]

is a

Krull domain. (1)

If

A is a nonzero integral ideal of D ,

(A[S])y = A y [S] (2)

. F -- ► FD[S]

The mapping

of

induces an injective homomorphism C(D[S])

If

S is a group, then

C(D[S]) = C(D) Proof. D

of

into

C (D)

F(D[S])

into

(1):

and

Letbe the set of minimal primes

that contain

A

and for

1 £ i £ k , let

K , the quotient field of

with the valuation ring k (Pits]}!

is surjective

.

normed valuation on

that

<J>

F (D)

.

(3)

of

then

V- = D p

.

v?

associated

D[S]

contains

A[S]

D[S]p [sj

, defined as in the statement of Theorem 15.3

have

Let

D,

be a

Corollary 15.9 shows

1 is the set of minimal primes of .

v^

that

be the valuation associated with We

212

A v = n£_1

^

, where

(A[S])y = n^(Pi [S])

,

nu = inf(v^fa) | a e A) , and where

n i = inf{v?(b)| b

It is clear from the definition each

i .

of

v*

that

e A[S]}

n^ = nu

for

Since

A yD[S] = (n^pfHi^DfS] to complete the proof of

(1)

= n^CP^n i ) [S])

(P^[S])^n i^

from part

(2) of Corollary 8.7:

and cancellative, the contraction to

for each

P?n i^[S] D[S]

,

, it suffices to show that

pf-n i^ [S] =

is

i .

This follows, however,

since

S

is torsion— free

P^[S]—primary , and hence is

of

P-n i ) [S] . D[ S]p.[s]= (Pj[S]D[S]p [s])n i This completes the proof of (2): [div(F

The mapping

• D[S])]

.

(1)

<J> is

Using

(1)

•

•

defined by

<J>([div(F) ])

, the proof that

(j)

=

is

well-

defined is the same

as the proof of the same assertion

a

Clearly

in Theorem 16.1.

[div(F)] F

e ker <J> .

Since

fD[S]

monomial, and since invertible in D



is a homomorphism.

for

Suppose

Without loss of generality we assume that

is an integral ideal of

principal.

to

.

D

FvD[S] = fD[S]

contains monomials,

Fv c aXsD[S]

S , and hence

then yields

. Then

Fy = aD

is

f = aXs

, it follows that

FvD[S] =aD[S] , so that

.

s

is

a

is

Contraction

[div(F)]

=

is injective, as asserted. (3):

Suppose

S

is agroup.

each nonzero element of if

N = D \ {0} , then

Theorem

S is oftype

(D[3])N

= K[S]

(0,0,...

15.6 shows that )

is factorial.

.

Hence, Thus

0and<J>

213 K[S]

has trivial class group, and Theorem 16.1 shows that

C(D[S])

is generated by

D [S]

that meets

that

D[S]

D} .

is a minimal prime of

Applying Corollary 15.9, it follows

is generated by

prime of of

N} .

{[div(Q)] | Q

{[divP^JS]] | P

Since each

<J) , we conclude that

[divPa [S]] <J>

is a minimal

belongs to the range

is an isomorphism of

C(D)

onto

C(D[S ]) . Suppose

D[S]

16.2 determines S .

is a Krull domain.

C(D[S])

in terms of

THEOREM 16.3.

S

and

S

of Theorem for a group

need not be a group.

Assume that

be the quotient field of

group of of

D

(3)

Using this result, Theorem 16.3 advances such a determi­

nation in the case where

K

Part

S , and let

H

D[S]

is a Krull domain.

D , let

G

Let

be the quotient

be the group of invertible elements

S . (1)

C(D [S] ) ~ C(D) © C(K [S]) .

(2)

As

in the statement of Theorem 15.2,express

H © T , where C(K[S])

T = G n F+ , with

~ C(L[T])

Proof,

, where

(1):

Let

L

D[S]

: C(D) -* C(D[S])

a homomorphism.

that the

defined by

Moreover, part

map y: CCD) -- ► C(D[G])

[divFD[G]]

is an isomorphism.

identity map

on

summand of

Since

as

Then K[H] .

be defined as in

D[G]

is a quotient

, Theorem 16.1 shows that the mapping

y C(D[G])

oi:C(D[S])

a free group.

is the quotient field of

the statement of Theorem 16.2. ring of

F

S

C(D)

C(D[S])

—

a([divF]) = [divFD[G]]

(3)

of Theorem 16.2 shows

defined by Since

y([divF]) =

y *a<J) is

, it follows that <J>(C(G)) say

is

C(D[S]) = <J>(C(D)) © M

the is a direct .

If

214 N = D \ {0} , then as in the proof of Theorem 16.2, there exists a surjective homomorphism with kernel generated by of

from

t

C(D[S])

([divP^fS]] | Pa

onto

C(K[S])

is a minimal prime

D) ; since this latter subgroup is precisely

it follows that

<J>(C(D)) ,

C(K[S]) ~ C (D [S ])/(C(D)) ~ M .

Therefore

C CD [S]) = <|)(C (D)) © M ~ C (D) © C(K [S]) , where the isomorphism C(D) a (C(D))

follows from part

completes the proof of part (2) :

We have

K[H]

K [S ] ~ K[H][T] .

This

, so part

(1)

shows that

But Theorems 14.15 and 15.1

is factorial, so

C(K[S]) ^ C(L[T])

of Theorem 16.2.

(1) .

C (K[S]) ~ C(K[H]) © C(L[T]) show that

(2)

C(K[H]) = {0} , and

, as asserted.

In view of Theorem 16.3, we are left with the problem of determining form

C(L[T])

n F+ , where

, where G^

L

is a field and

T

is of the

is the quotient group of

handle this problem in two stages.

T

.

We

First, we define the

v— operation on an arbitrary cancellative monoid

S .

For

S

completely integrally closed, we define the divisor class group

C(S)

of

S .

This development is analogous to the

usual treatment for integral domains, and hence will provide a review of the material concerning divisorial ideals, class groups, etc. in domains that comprises most of the intro­ duction to this section.

If

K

is a field and

Krull domain, we prove in Theorem 16.6 that

K[S]

C(K[S]) ~ C(S)

This result can be viewed as the counterpart of part Theorem 16.2.

Then for

Theorem 16.8 that

S

of the form

C(S) ^ F/G .

Krull, that we seek.

(3)

. of

G n F+ , we prove in

When combined with Theorem

16.3, these results provide the description of D[S]

is a

C(D[S])

, for

To begin, we develop the theory

215 of the

v—operation and related concepts for a cancellative

monoid. Let

S

be a cancellative monoid with quotient group

A nonempty subset S if that

(1)

I

of

S+ I c I

s + I cs .

G

and

is called a fractional ideal of (2)

need not be a subsemigroup of

by

F(S)

ment {x

subsets of

b e B) S

G .

.

— If

e G |x

s € S

A fractional ideal

I = x + S

for some

the set of fractional ideals of

addition of and

there exists

We remark that a fractional ideal

said to be principal if

F(S) I ,J e

G—

that is,

x € G .

S .

A + B

such of

S

I

is

Denote

Under ordinary

= ,{a + b | a e A

is a commutative monoid with zero ele­

F(S)

+ J £ 1}

, then

I :J

.

is defined to be

We show in part

I:J c F(S)

by

Iv and is called the divisorial ideal associated with I =I

, then I

THEOREM 16.4.

The fractional ideal

S:(S:I)

(1)of Theorem 16.

that

if

.

G .

is denoted I;

is divisorial. Let the notation and hypothesis be as in

the preceding paragraph. (1) (2)

If I,J 6 F(S)

, then

I:J € F(S)

I:(x + J) = - x + Cl:J)

particualr,

S:(S:(x +S)) = S :(—

.

for each x + S) = x

xe G . + S , so

In x+ S

is divisorial. (3)

If

E J 2 > then

I:J^ 2. I:J2 •

Hence

CJi)y £ (J2^v * (4)

Iv is

the intersection of the family of all prin­

cipal fractional ideals of

S

that contain

I .

216 Proof.

C D : Take

L n S * <J> and that

(L

n S)+ S £

is clear.

For the first, take

where

e S .Then

u,v

u e L n S . and so

To show that such that

u + a e

I:J .

so

+I E S .

b +

omitted. I v

That S + (I:J)

- x e S :I

+S 2

and

I

and take

b e S

b + v + xeb + Is

I £

I

such

s,

x +

S

principal fractional ideals

for some

s e S .

for some

but

y - x ^

S , so

follows

if and only if

x

(1)

.

implies

is contained in the interv

^ x + S

(5)

9

(3)are routine, and hence are

y

Statement

n S

, and this completes the proof of

(2)

- s + S

hence

is clear from the

I ; this family is nonempty since

I £

is such that

x

nS

Ifx e I :J , then

section of the family of

implies

and

as u — v ,

u + a + J cu +S ci

c I:J

v e J

= x + S . so *

that contain

and write it

is nonempty, choose u e I

To prove (4) , note that

£ (x + S) v

. Wefirst observe that

The second assertion

a + J eS . Then

v + (I :J) £ S

The proofs of

L.

y e L

I :J

definitions. Finally, take b

F(S)

u = v + y e S + L £ L ,

a e s

that

L e

of S

s +I

£ S

Conversely, if

x + S

containing

y

e G

I, then

y { S:(S:I) = Iy .

from

(4)

and

from the fact that

+ S 2 Iy . To

prove

(6) , we

note that Iv = n{a +

S | I £ a + S} , and hence

x +

+ n{a + S} = n{x + a + S | I £ a + S } =

Iv = x

n(x + a + S | x + I c x

+ a+S}

n{b + S | x + I c b + S } = ( x + (7)The inclusion (I

+

J)y£

(Iy

I + J £ I+ + Jy )v .Moreover,

= 1)^ implies that since

Iy + Jy£(I

+J)y ,

217 then

(Iy + Jy )v c ((I + J)v )y = (I + J)y , and hence the

equality If

S

then the F(S)

(I + J)^ = (1^ + Jy )y

is a cancellative monoid with quotient group

v—operation induces an equivalence relation

defined by

div(I) P(S)

holds, as asserted.

I ~ J

if

I

= Jy .

For

denotes the equivalence class of

under

denotes the set of all divisor classes of

of Theorem 16.4 shows that the operation fined by Under

divfl) + div(J) = divfl + J)

+ , the set

zero element

P(S)

div(S)

.

~

I € F(S)

I

+

,

on

,

~ , and

S .

on

G

Part

P(S)

(7)

de­

is well-defined.

forms a commutative monoid with Moreover, the set

P(S) =

{div(x + S) | x e G}

is a subgroup of the group of invertible

elements of

The factor monoid

PCS)

.

called the divisor class monoid of integral domains, of

C(S)

.

[div(I)]

C(S) = P(S)/P(S)

S .

As in the case of

denotes the element

Under what conditions is

PCS)

div(I) + P(S)

a group?

Based

on the domain

case,the answer should be "if and only if

is completely

integrally closed".

S

some fixed element THEOREM 16.5. quotient group

G .

Recall from

is completely integrally closed if

contains each element

x

of

s e S Let

G

S

Theorem 16.5 shows that,

indeed, this is the appropriate condition. Section 12 that

is

such that

s + nx e S

S for

and for each positive integer S

n

.

be a cancellative monoid with

Then

P(S)

is a group if and only if

S

is completely integrally closed. Proof. To show that

Assume that P(S)

an inverse in P(S)

S

is completely integrally closed.

is a group, we for each

I e

show that F(S)

.

[div(I)]

has

Without loss of

218 generality we assume that in

S .

I

is a divisorial ideal contained

It suffices to show that

that is, that

(I +

(S:I))v = S .

i t ’s enough to show

that

fractional ideal

+ S

x

S of

Therefore

I e - x +I e I

over

S , so

- x

almost integral

n e z+ , where U n = 0 (ny

+

in

P(S)

so

y e S

S)

.

s e S . of

S

Since and

Suppose

over

K

•

S

- x

Then

1 +

because

field of

is a field and C(K[S])

K[I]

1

=

Let

y e G

for each

1

’

so

d

i

v

*

it follows that

K[S]

divfS)

I £ S

and

is a Krull domain.

C(S)

In

are isomorphic, one

The statement of Theorem 16.5 for a fractional ideal

for the family of elements

I need not itself be a semigroup;

I

f e K [G ]

denotes the set of elements K[S]

i t ’s easy to show,

K [I ] is a fractional ideal of K [S]

such that

THEOREM 16.6.

yK[J]

Assume that

K [I ] :K [J ] = K [I :J ] for

y

c K[I]

S

. As usual,

of the quotient

.

is a torsion— free can­

cellative monoid with quotient group (1)

is almost integral

Supp(f)c I . This is a variance from usual notation

however, that K[I]:K[J]

for

be the fractional ideal

contains an aberration in notation:

such that

. Thus

S is completely integrally closed.

preliminary result is needed.

S , we write

- nx + I £ I

s + ny e S

say I

I + (S:I)

is a group.

is divisorial,

order to prove that

of

S ,

+ S , as asserted.

P(S)

S — Let

+ (S:I) e

I c S :(S:I) = Iy = I .

, and

and S E x

Conversely, assume that be

-x +

n . Hence

e S

I

S containing

impliesthat

each positive integer

Since

is contained in each principal

x + S ^ I + (S:I) -2x +

div(I + (S:I)) = div(S) —

G .

Let

I ,J £ F (S) .

K

be a field.

219 (2)

(K [I ])v = K [Iv ]

(3)

The mapping

y: P(S) --- P(K[S])

y(div(I)) = div(K[I]) .Proof. mediate. s e

J

(1):

.

is

The inclusion

yXs £ K[I]

y e K [G ] , however, yX S e K [I ]

K [I :J ] sK[I]:K[J]

implies

take

y

and each

s e J

iff

s € J

y £ X'SK[I] £ K[G]

t + s e I

iff

is im­

e K [I]:K [J ] .

it is clear thaty e K [I ] :K [J ]

for each

t e Supp(y)

by

an injective homomorphism.

For the reverse inclusion,

, then

defined

If

. For

iff

for each

Supp(y) E I :J

iff

y € K [I :J ] . (2)

It follows from

(1)

that

(K[I])y =

K [ S ]:(K[S]:K [I]) = K[S:(S:I)] = K[Iy ] . (3): Iy = J y

For iff

I ,J e F(S)

, we have

K[Iv ] = K[Jy ]

div(K[I]) = div(K[J]) and injective.

.

iff

(K[I])y = (K[J])y

Therefore

Moreover,

div(I) = div(J)

y

iff iff

is both well-defined

y(div(I) + div(J)) = y(div(I+J)) =

div(K[I + J]) = div(K[I] • K[J]) = div(K[I]) + div(K[J]) = yCdiv(I)) + y(div(J )) , so

y

THEOREM 16.7. Assume that a Krull domain.

is a homomorphism. K

is

$ :C (S)

The mapping

d>C [div(I) ]) = [div(K[I])]

a field and -► C(K[SJ)

K[S] defined

is an isomorphism of C CS)

is by

onto

C(K[S ]) . Proof. (3)

Let

y:P(S) -- ► P(K[S])

of Theorem 16.6.

div(XyK[S])

Since

<J>.

injective and surjective. is principal and

K [Iv ]

part

y(div(y + S)) * div(K[y + S]) =

, it follows that

duces the homomorphism

be defined as in

p(P(S)) s P(K[S]) , so

We show that If [div(I)]

<J>

is

in-

both

e ker <j> ,then

contains monomials.

p

Therefore

K [Iv ]

220 K [Iv ]

is generated by

Xa

for some

K [Iv ] = XaK[S] = K[a + S] so

[div(I)] = 0

and



a e G .

implies, however, that

plies that

K[G] = K[S]N , where K[G]

is a Krull

domain,

C(K[G]) = {0}

shows that

is generated by

minimal prime of

K[S]

Qa

{[div(Qa )] | Qa

Ia of

<J>

a ,

is a

Consulting Corollary

is of the form

[div(K[Ia ]) ] = ()>([div(Ia ) ])for each , and hence

is surjective.

Therefore Theorem 16.1

N) .

an appropriate proper prime ideal

C(K[S])

4>

and then Theorems 15.1

.

that meets

15.9, we conclude that each

range 4> 2

= a+S,

N = {Xs | s e S} ,im­

and 14.15 show that C(K[S])

I

is injective.

To complete the proof, we show that The equality

The equality

K [Ia ]

S .

for

Since

it follows that

is surjective.

This com­

pletes the proof of Theorem 16.7. A combination of Theorems 16.3 and 16.7 yields the following corollary. COROLLARY 16.8. domain, then

C(D[S]) ~ C(D)

Assume that

S

and assume that of

S .

Let

If the monoid domain

H = {0}

{W } » a aeA

7

G

that are essential for

Pa

on

be the center of

{e } » a ae A

wa

associated with S .Theorem

F =

’ ™ ^ ere

family

a:G -- ► F

S , let

w^

Wa , and let

15.8 shows that

be the weak direct sum of the Za = Z

^or each

be the canonical free basis for

the mapping

G

be the family of rank-one discrete

G

Let

© C(S) .

is the set of invertible elements

be a normed valuation on

.

a Krull

is a Krull monoid with quotient group

valuation monoids on

S = nae^Wa

D[S] is

defined by

a » anc* let F .

Since

a (x) = Eawa (x)ea

H = {0} , is an

221 imbedding of

G

in F . Part

that

=S

in this case, and

G n F+

(3)

ofTheorem 15.2 it is an

S

shows

of this form

to which Theorem 16.3 reduces the problem of determining the divisor class group of a Krull monoid domain. THEOREM 16.9.

Let the notation and hypothesis be as in

the preceding paragraph. (1)

*S a ^ree basis for

(2)

Let

set of {wa } value on Then

y € G\{0}

and let

i

div(y + S) = divCE^w i(y) C(S)

- F/G

each of

(1)

and

Krull domain

(3)

{div(K[Pa ])} (divfP^)}

*

if

w^(y) < 0 .

(2) , In proving K[S]

K , and we transfer properties of the

K[S]

That

S .

(2) , we consider the monoid domain

over a field

from part

on

l

We remark that in the statement of

means -w^(y)(S:P^)

(1):

w.

.

Wi(y)Pi

S

be the finite sub­

P. for the center of

7

Proof.

of

.

consisting of those valuations with nonzero

y .Write

(3)

P(S)

to

S .

(div(Pa )}

is a free subset of

P(S)

follows

of Theorem 16.6 and the fact that is a free subset of p(K[S])

generates

P(S)

To

prove that

, it suffices to show that

belongs to the subgroup generated by tegral divisorial ideal

.

I * S

of

(div(Pa)} S .

Part

div(I)

for each in ­ (2)

of

Theorem 16.6 shows that

K [I] is aproper divisorial ideal

of

15.9shows

K[S]

primes of (Qj>

T

, and Corollary K[S]

containing

*s t*10 subset of

that contain

I .

Hence

{ pa }

K [I ]

that the set is

of minimal

(K[Q^.]}?=1 , where

consisting of elements

div(K[I])

P^

belongs to the subgroup

222 of

P(K[S])

generated by

{div(K[Q^])}™ , and since each of

these elements is in the range of the mapping 16.6, it follows that P(S)

generated by (2):

(2) , it suffices to consider the case

K[y + S] = X^K[S]

15.7.

(3):

Let

w^

div(y + S)

jective. ^lm iei

(1)

as in the proof of Theorem

a(G)

According to parts

i

and

But this is precisely the condition that kerT

= a(G)

as asserted, and

Thus, if

(DtkP^^,

is

(1) , however, this

y e G\{0}

w a (y) = 0

.

=

if and only if

occurs if and only if there exists for each

t

F , then

and

is sur­

t

is the kernel of

(2)

F

determined by

implies that

Znu [div(P^)] = [divCEm^^)] = 0

w^(y) = nu

is the valuation on

t:F -- * C(S)

a nonzero element of

principal.

is deter­

canonical free basis for

Part

We show that

w*

k^

= div(I^w^(y) P^), as asserted.

and consider the homomorphism T (ea) = [div ( Pa) ] .

Moreover, since

the integer

, where

^e a ^aeA

div( y + S) =

in this case.

associated with

Therefore

shows that there exist

such that

is principal,

w*(Xy ) = w^(y)

q.f.(K[S])

(1)

k^,...,k

Elkidiv(Pi) = div(EikiPi)

mined as

belongs to the subgroup of

The proof of

positive integers

of Theorem

(divCQj)}™ .

To prove

y e S .

where

div(I)

y

for

such that a J {l,...,n} .

Em^e^ = o(y)

C(S) ^ F/a(G)

c* F/G .

. Thus This

completes the proof of Theorem 16.9. Since Theorem 16.9 provides an alternate description of C(S)

in the case of a Krull monoid

vertible element,

S

with no nonzero in­

its combination with Theorems 16.3 and 16.7

yields the following result.

223 COROLLARY 16.10. Let

H

press let G

Assume that

D[S]

is a Krull domain.

be the group of invertible elements of S

as

H © T .

Let

G

S , and e x ­

be the quotient group of

{wa^aeA

T ,

rank-one discrete valuations on

that are essential for

direct sum of groups imbedding of

G

in

T , let

F =

Za = Z , and let F .

Then

be the weak a

C(D[S])

be the canonical C(D) © (F/a(G)) .

We conclude the section with a result concerning Krull monoids that follows from Theorem 16.9; the statement of Theorem 16.11 is an expected result. THEOREM 16.11. group

^ QL^0Lef^

G , let

valuation monoids on Pa

Let

S

be a Krull monoid with quotient

be the family of rank-one discrete

G

that

be the center of Wa

on

of minimal prime ideals of Proof.

Clearly each

S , and

let

S .Then

is

set

^pa ^aeA

t*10

S . Pa

noinclusion relation between Thus, to prove

are essential for

is prime in

S , and there is

Pa and

for

Pg

a * 3 .

(16.11), it's enough to show that each prime

ideal

P

of

S

contains some

where

H

is the group of invertible elements of

definition, each

Pa

Pa .

is contained in

straightforward to show that

Express

S

as S .

H © T , By

T , and in fact, it's

is the family of centers

of rank-one discrete valuation monoids on the quotient group of

T

P

by

that

that are essential for

T , a Krull monoid.

Replacing

P n T , we therefore assume without loss of generality S = T .

Choose

x e P .

Part

(1)

implies that there exists a finite subset and positive integers

of Theorem 16.9 ^P^}^

such that

of

(Pa )

224 x + S = hence

. P

Therefore

contains some

P ^ x + S 2 ^k^P^

, and

P. . 1

Section 16 Remarks Detailed treatments of the material cited in the intro­ duction to this section concerning the

v—operation on a

Krull domain can be found in [20; Ch. VII, Sect. Ch.

I and II], and [51; Sect.

34, 44].

1],

[44;

In particular,

is a special case of Theorem 7.1 of [44].

(16.1)

Other references

for the divisor class group of a Krull monoid domain are [95],

[7],

[28], and [21].

More specifically, part

(16.2) is Proposition 5.3 of [95], part is part

(1)

of

of

of (16.3)

of Proposition 7.3 of [7], and Corollaries 16.8

and 16.10 appear in [28]. culations of

(1)

(3)

CCD[S]) , see

For some specific examples of cal­ [6— 8 ], [28], and [21].

\

CHAPTER IV RING-THEORETIC PROPERTIES OF MONOID RINGS

Chapter IV continues the main theme of Chapter III — that of determining,

for various ring— theoretic properties

necessary and sufficient conditions on a unitary ring a monoid

S

in order that

R[S]

R

should have property

E , and

E .

The main difference in the two chapters is that in Chapter IV, R[S]

is not normally assumed to be a domain; hence

contain zero divisors and

S

R

may

is not uniformly taken to be

torsion— free and cancellative.

In Section 17,

E

is the pr o ­

perty of being (von Neumann) regular, and certain chain conditions,

including

d.c.c.

, are considered in Section 20.

It is trivial to determine conditions under which a monoid d o ­ main is regular or Artinian (that is, a field). hand, the properties

On the other

E considered in Sections 18 and 19, in­

cluding the conditions of being a Priifer ring or an arithmet­ ical ring, are closely related to properties considered in Section 13 of Chapter III.

One new wrinkle arises in the case

of rings with zero divisors, however.

While a unitary inte­

gral domain is arithmetical if and only if it is a Priifer domain, in Section 19 we show that if then

R[S]

S

is not torsion— free,

may be arithmetical, but not a Priifer ring.

Similar situations exist in regard to the properties Bezout ring-arithmetical ring and the properties of being a princi­ pal ideal ring or a

ZPI— ring. 225

226

§17.

Monoid Rings as von Neumann Regular Rings

Throughout this section we use the term regular ring for a ring that is regular in the sense of von Neumann. inition, for that each x e R .

a ring

a e R

R

The def­

that need not be commutative, is

isexpressible in the form

For

R

commutative,

R

pal ideal of

R

is idempotent iff

axa

for

some

is regular iff each princi­ R

is a zero— dimensional

reduced ring. We assume throughout the section that (commutative) ring and that

S

is a monoid.

results are directed toward a proof that mensional if and only if periodic.

Since

inequality

R

dim R £ dim

torsion— free rank = a

is a unitary Our first

R [S ]

R [S ]

Assume that a , then

is zero— d i ­

is zero— dimensional and

is a homomorphic image of

THEOREM 17.1.

|A|

R

R

R[S]

S

is

,the

is always satisfied. G

is a group.

If

G

dimR[G] = d i m R [ { X . K A ] A AeA

has , where

.

Proof.

First, a comment on the statement of the theorem

seems appropriate.

We do not distinguish among different

infinite cardinalities in dealing with Krull dimension, so if

a

is infinite, the equality in the statement of (17.1)

is interpreted as meaning that Let

R[G]

be a maximal free subset of

H =

he the subgroup of

Then

G/H

is a torsion group

so

R[H]

and

dimR[G] * dimR[H]

.

rank of

is infinite-dimensional.

H

ality that

is

G

generated by R [G ]

G , and let {y^)

•

is integral over

Moreover,

the torsion— free

|I| = a , so we assume without loss of gener­

G = H .

In that case,

R [G ]

is isomorphic to

227

R [( \ >A eA *( \ R[(X

A

*

T ^e latter ring is integral over

+ X ^} ] , for each of

X

A

monic polynomial

Y 2 - (X

and

A

X *

+ X *)Y + 1

A

is a root of the

A

over

A

R[{X

+ X*}]

A

.

A

A straightforward degree argument shows, however, that {X

+ X *}

A

is algebraically independent over

A

quently,

dimR[G] = dim R [{ X ^

COROLLARY 17.2.

If

is periodic, where Proof.

~

^

dim R £dimR[S/~]

since

R[S]

dimR[S/~] = n

group of R[G]

S/~ .

Then

and

S/~

Proof.

dimR

.

Let

R[S/~]

S .

G

.

be the quotient since

The equality

Consequently,

G

of

dimR=

G

has

is a torsion group

is periodic.

dimR[S] = k

and

is a homomorphic image

dim R £ dimR[G] £ dimR[S/~]

0 .

THEOREM 17.3. Then

S/~


then implies, by Theorem 17.1, that

torsion— free rank

R

R[S/~]

is a quotient ring of

dimR[G] - n

, as asserted.

is the cancellative congruence on

We have

Therefore

Conse-

dim R = dimR[S] = n < °° , then

inequality holding .

R .

If

Assume that

S

and choose

Thus

ms + c * ns

{t

e

I

is an ideal of

if

and only if I

S | k^s

+ t=

S

is periodic, =dim R . s e S .

exist distinct positive ns , where

has finite dimension

if and only if

dimR[S]

ms ~

R

is periodic.

then

R[S] is integral over

Conversely,

assume that

integers

for some

k 2s + t S , and

dimR[Sj =

Corollary 17.2 shows that there m

and

n

such that

~is the cancellative congruence + c

k .

c e S .

for some s

Let

on

S

I =

k^,k2 e z+ >

*

k 2) \

is a periodic element of

meets the semigroup

< s> of

.

S .

S

Assume

228 that

I n <s> = <J> .

ideal U

J

of

S

By Theorem 11.8, there exists a prime

containing

be the submonoid

R[ J]

S\J

is an ideal of

quently, so that

I

of

R[S]

such that

S .

, and

R [ U ] ^ R[S]/R[J]

Then R[U]

J n <s> = <J> . R[S] = R[U]

n R[J]

k = dim R < dimR[U] < dimR[S] = k , and

U/~

is periodic.

k^s + u = k 2s + u k2

Since

u e U .

contradicts the fact that periodic and

S

and

R [S ]

R[U]

, we see

But then

k^

and

u e J n U , which Consequently,

s

is

is periodic, as asserted.

R is reduced,S

pis regular on

p— torsion— free.

Conse­

s e <s> c U , it follows that

J n U = <J> .

Theorem 9.17 shows that only if

.

dimR[U] =? k .

for distinct positive integers

and some element

+ R [J ] ,

is a homomorphic image of

Applying Corollary 17.2 to the monoid ring that

= (0)

Let

R

R[S]

is a reduced ring if and

is free of asymptotic

for each

p

such that

torsion, S

is not

This result and Theorem 17.3 enable us to

determine the monoid rings that are regular. THEOREM 17.4. a monoid.

Let

The ring

R

R[S]

be a unitary ring and let

S

be

is regular if and only if the

following four conditions are satisfied.

S

(1)

R

is regular.

(2)

S

is free of asymptotic torsion.

(3)

p

is regular on

is not (4)

for each prime

p

such that

p— torsion— free. S

Proof. R

R

is periodic. If

R[S]

is regular,

is a homomorphic image of

and Theorem 17.3 implies that

R[S] S

then .

R

Thus

is regular since dimR=dimR[S] =0 ,

is periodic.

Moreover,

229

R[S]

reduced implies that

Therefore conditions regular.

(2)

(1)— (4)

and

(3)

are satisfied.

are satisfied if

invertible, condition

(3)

the condition that

is a unit of

such that

S

p

is not

tic of

for each

and only if

R

R

and if

p

A e A , then

p

p J {p } . A A €A

>

for each prime

Thus,

If

{M_}_ , A AeA

A

is a unit of

R

if

stated in terms of

A , where

p^ * 0 , then condition

in particular,

S

is

"0— torsion— free " is in-

(3)

Note that if

implies condition

this is the case if

R

(2)

;

has nonzero character­

To obtain other equivalent forms of the conditions in

Theorem 17.4, we consider some consequences of nation of conditions THEOREM 17.5.

(2)

and



element

Moreover,

t

of

T .

(4)

if

is periodic, then

T

prime

p if and only if

p

As in Section 2, let

and index, respectively, of

nition of

If

r > 1 , then r .

k(m + r - l)t

But for since

is free of asymp­

T

divides the order of no element of

r = 1 .

T

is a group for each periodic

p— torsion— free for a given

Proof.

the combi­

of that result.

If the semigroup

totic torsion, then

is

is the

is the characteris-

terpreted as "free of asymptotic torsion".

istic.

p

can be replaced by the condition that

p — torsion— free for each

some

regular ring are

of Theorem 17.4 is equivalent to

p— torsion— free.

set of maximal ideals of

A A €A

is

The converse follows by similar reasoning.

Since regular elements of a von Neumann

{Pi}-* a

R[S]

t .

T . m

and

r

be the period

It suffices to prove that

(r - l)t *

(m + r - l)t

by defi­

k > 2 , we have k(r - l)t =

k(r - 1) = k(m + r - 1) (modm)

and

230 since

k(r — 1) ^ r .

This contradiction to the fact that

is free of asymptotic torsion shows that



T

is a group, as

asserted. Since any two elements of a it follows that if no nontrivial ible by

T

is

p

Conversely, if

T

is not

contains

generate the same subgroup of ments of the subgroup G since

and b

such that

c , then

T ; hence

G = d

p— torsion— free, then

b,c e T

does not divide the order of

b + d * e

T

p— group, and hence no element of order divis­

p .

identity of

p—equivalent,

p— tors ion— free, then

there exist distinct elements If

p— group are

of

T .

b If

is the inverse of

c

and

and e

c

pc are e l e ­

is the

c

in

* c , but p(b + d) = e .

divides the order of an element of

pb = pc .

G , then Therefore, p

S .

In view of Theorem 17.5 and the remarks following Theorem 17.4, we can state a variety of conditions on S

that are equivalent to the condition that

R[S]

R

and

is reg­

ular . COROLLARY 17.6. is the set char(R/M^) = p^

Assume that

R

of maximal

for each

A e A

is regular,

that

ideals of .

R ,and that

Assume that

odic and free of asymptotic torsion.

S

is peri ­

The following conditions

are equivalent. (1) is not

p

R for each prime

p

such that

p— torsion— free.

(2) the order (3)

is regular on

p

is a unit of

of an element of Nop^

divides the

R for each

prime p

that

divides

S. order ofan element of

S .

S

231 If the semigroup

T

is periodic and free of asymptotic

torsion, then Theorem 17.5 implies that periodic subgroups.

T

is a union of

We embark on a brief consideration of

this and some related conditions in a semigroup

T

.

Thus,

the material in the rest of the section is primarily con­ cerned with some of the theory of commutative semigroups that was not covered in Chapter 1.

In particular, we abandon for

the rest of the section the restriction to consideration of monoids that has been the rule for several sections.

We use

the term semilattice to describe a semigroup, each of whose elements is idempotent.

The results concerning the

Archimedean decomposition of a semigroup established in Theorems 17.9 and 17.10 will subsequently be used in Section 23. THEOREM 17.7. of a family

Assume that the semigroup

^ a ^a€A

subgroups of

semilattice of idempotents of u{Ga | e

subgroup of and

H

e

G,

periodic, then each If

for Hg

idempotent of e^ € t + T . t + e = t (e d

T

with

To prove

and e = t + y

+t)+y = e + Ol

E

be the

e E , let Each

Hg

Hfi = is a

Moreover,

if each

G , G„

is

a

is also periodic.

t e T , then

t + x = e . a

is the union

a partition of

e,f e E .

be the identity element of such that

Ga > .

the family

+ Hr E H ~ f e+f

Proof.

T . Let

T , and for e

is the identity element for

T

t e Ga

Ga .

for some

Then there exists

We observe that

e a

the properties that t this, assume that for

(t+y)®e CX

a .

some

y +

€T

e

Let

ea

x e Ga

is the unique n + e^ = t

e

eE

.

Then

and issuch that

e = t+ y

= (x +

t)+

= ol

e = x + (t +

232 Thus, and

He

may be alternately described as

e e t + T } .

Since each

potent and since T= uG^ is a partition of T , take

a,b e

closed.

G.

He .

and e = e + e e

, it

G

a

contains a unique idemn

follows

To prove that

Then

(a+b)

(a + T) + (b + T)

It is clear that

{te T | t + e = t

e

easily that Hg

is a subgroup of

+ e = a +

(b + e) = a +

<=a + b + T

; hence

b ,

He

is

is the identity element of

t e G c H , there exists s e G such that a e a t + s = e ; since s e He , it follows that each H e

He ,

and if

subgroup of

T .

a e He , b e f e b + T e + f e

.

Then

a = a + e , b = b + f , e e

- , and hence e+f

a + T ,

a + b = a + b + e + f

+ (b + T) c H

a + b +T .

This

+ H,- c H ~ , as f ” e+f *

e

nally, it is clear that each

H

J

a

£ He+f 9 ta^e

prove that

. Consequently,

(a + T)

a + b e H

To

is

e

and

and

implies that asserted.

is periodic if each v

FiG

a

is

periodic. COROLLARY 17.8. in a semigroup

The following conditions are equivalent

T .

(1)

T is periodic and is free of asymptotic

(2)

T is a union of periodic subgroups.

(3)

T is a disjoint union of periodic subgroups.

Proof. prove that uacA^a

In view of Theorems 17.5 and 17.7, it suffices to (3)

implies

a partiti-011

It is clear that

T

=3

since

<s> c G

(1) G

.

Thus, assume that

T =

into periodic subgroups

is periodic.

asymptotically equivalent. a

torsion.

If

Assume that

s e G

a and

and

s,t e T

t e G

, c G„ , <s> n a 3 Choose relatively prime positive integers n and

3

G^ . are

, then * d>

m such

.

233 that

ns = nt

tegers

x

and

and

ms = mt

y .

, and write

Since

1 = nx + my

is a group, we have

nxs + mys = xnt + ymt = t .

Therefore

T

for in­ s =

is free of asymp­

totic torsion. Assume that

T

is a periodic semigroup that is free of

asymptotic torsion. to

T , where

is the family of cyclic subgroups of

T , then elements of

We remark that if Theorem 17.7 is applied

T

a ,b e T

belong to the same subgroup

if and only if the identity element for

same as the identity element for



He

is the

< b> .

Theorem 17.7 is a special case of a more general decom­ position theorem for commutative semigroups that we proceed to establish.

To partially motivate the development, we note

that in Theorem 17.7, the set

E

of idempotents of

semilattice and that the partition with the operation on for all

e,f e E .

E

T = ueegHe

in the sense that

U = u

U + U c U a p a +p

is a

is compatible

Hg +

c ^e + f

In general, we say that a semigroup

a semilattice of subsemigroups semilattice,

T

AU aeA a

for all

is a

{ua |a ^

e A} if A

is

partition of U ,

a ,3 e A .

U

is

a and

The decomposition theorem

referred to above states that each commutative semigroup

T

admits a unique decomposition as a semilattice of Archimedean subsemigroups, where a semigroup

U

rad(x + U) = rad(y + U)

x,y e U .

for all

is Archimedean if (The terminology

Archimedean is suggested by the theory of ordered groups, for such a group is Archimedean, as customarily defined,

if and

only if its semigroup of positive elements is Archimedean as defined above.)

To obtain the

decomposition theorem,we

introduce a new congruence on a semigroup

T

.

234 THEOREM 17.9. on

T

by

s p t

(1)

p

U = T/p

if

is a congruence on

T/y

(2)

Let

under

T , define a relation

rad(s + T) = rad(t

is a semilattice.

such that

T

For a semigroup

.

T , andthe factor semigroup

If

y

is a semilattice, ^ u ^u€u

+ T)

p

is any congruence on then

T

y > p .

be the set of equivalence classes of

p , indexed by the elements of under the

U

canonical

in such a way

that

Tu maps to

u

map f:T

Then

T = u ueuTu

is the unique representation of

-► U

.

T as

a

semilattice of Archimedean semigroups. Proof. is clear.

(1):

To prove that

operation

on

ns e t + T

That

p

n e

t + x + T .

s p t

Z+ , so

and that

n(s +

s + x e

Consequently,

is an equivalence relation on

is compatible with the semigroup

T , assume that

forsome

p

radft + x + T)

.

radft +

x + T)

c

rad(s + x + T)

so that

We have

t p

2t

Assume

that T/y

such that s + y

,so

and

and

some

(s +

t)y (2s + y ) y (s + y)y

t.

To prove Tu of

y>

p.

is a semilattice. s,t

e

ms = t + x

t y nt y(s + y) s and

t and

.

T and

are nt =

x,y € T . Thus

t)y(2t + x)y(t + x)y

s y

, and hence

(s + x)p (t +x)

T/p

(s +

Therefore

class

+ x)

Z+

s

Similarly,

This implies that

for some m,n €

Then

x) e t +nx + T

is a semilattice and that

s p t .

s y ms y(t

te T

e T .

rad(t + x + T)

rad(s + x + T) e

for each

x

, so that

This establishes

(1)

(2) , we first show that each equivalence T under p

is an Archimedean subsemigroup

.

2 35 of

T .

Thus, take

2a y(a + b) Therefore

.

Moreover,

Since

a y 2a

a + b e Ty ,

To show that x,y e T

a,b e Tu .

and

a y b , then

and hence

Tu

ma

= b + x

.

is a subsemigroup of

Tuis Archimedean, choose

such that

a y(a + b)

and

m,n

T .

e Z+

and

nb

=

a +y .

Then

(m + l)a = b + a

+ x ,where

a e rad(a

+

x +T) £ r a d ( a + T)

It follows that

rad(a + T) = rad(a + x

+

T), and

a + x e Tu .

The equation

implies that

a

+ l)a = b + (a + x)

belongs to the radical of

by the same argument, Therefore

(m

b

belongs to

Archimedean.

Clearly 7

T = u fTT ue U u

rad(a + Ty )

only show that

f(a + b) = u +

v

Wc

a,b e W c

is Archimedean,

it is

°f

T

= v , and

, b e W, , x e W_ , and d e ’

p b

clear that a p b

if

and that

c,d,e,f e C

y e VI ~ . J f

Then

and

b + x e W (j+e , so

c = d + e and hence

d +

e = c .Considering the equation nb = a

clude by similar reasoning that and

a

and b

it

a

nb = a + y .

c

as a semilattice

if and only if

b + x

Choose

T = uUeyTu ,

a,b e T ,

. For

Conversely, assume that a p b

a e w

T , and

for each a e Ty, be Ty .

a,b c W c . and

is

Tu + T v - Tu+v * we neec*

T = uce^ c

of Archimedean subsemigroups

Wc

Tu .

.

take any representation

Since

in

Ty

To prove uniqueness of the representation

suffices to show that

Tu ;

r

This is clear, however, since f(a) = u , f(b) f (a + b) = f(a) + f(b)

in

is a partition of

the indexing is such that to show

hence

then

b + Tu

r a d ( a + T ) = rad(b + Ty ) , so each

.

c+ d = d .

belong to the same

This completes the proof of Theorem 17.9.

set

ma

.

=

such that

ma e W

me

= W

c

c + d = 2d + e = + y , we con­ Hence Wc

c = d ,

of the partition.

236 With the notation as in Theorem 17.9, the semigroups are called the Archimedean components of

T.

is Archimedean,

of G .

for

G

is the only ideal

Any

Tu

group

G

Thus,

uniqueness of the Archimedean components implies that the subgroups

H0

of

T

in the statement of Theorem 17.7 are

the Archimedean components of

T .

It is in this sense that

Theorem 17.7 is a special case of (17.9).

For

T

free of

asymptotic torsion, Corollary 17.11 shows that a given Archimedean component

U

of

T

is a group if and only if

U

contains an idempotent. THEOREM 17.10.

The Archimedean components of

cancellative if and only if Proof.

T

If

a,b e T

then it is clear that

is free of asymptotic torsion.

such that

radfa + T) = rad(b

na = nb

and

na + a = nb + b = na + b

.

follows that

T

a = b , so

a,b,c

ponent

such

U of

T

such that both and

na

and

0 < i < k , then we have

U

, of

so

a

T

and

.

b

Choose Then

U is cancellative,

it

T

is free of asymptotic

are elements of an Archimedean com­ that a + c = b nb

; hence

na + b + b = na + 2b .

are

is free of asymptotic torsion.

+ c . Choose

belong to

nb = c + y . Then

b + c + x = n a + b

+T)

(n + l)a = (n + l)b .

Since

Conversely, assume that torsion and that

T

are asymptotically equivalent,

belong to the same Archimedean component

c + x

are

Assume that the Archimedean components of

cancellative.

n e Z+

T

c + U , say

ne

Z+

na =

(n+ l ) a = a + c + x =

(n + 2)a = (n + l)a + b =

By induction,

it follows that if

(n + k)a = (n + i)a +(k — i)b .Similarly,

(n + k)b = (n + i)b + (k - i)b

for 0 < i

< k.

In

237 particular, 2na = na + nb = Therefore a = b .

a

2nb

and

b

and

are asymptotically equivalent, and

It follows that

COROLLARY 17.11. torsion and that Then

U

U

U

is cancellative, as asserted.

Assume that

T

element for rad(a + U) vertible in

U

T .

contains an idempotent.

We need only show that the existence of an idem-

e e U

implies that

is free of asymptotic

is an Archimedean component of

is a group if and only if

Proof. potent

(2n + l)a = (n + l)a + nb = (2n + l)b .

implies that U U .

is a group.

Theorem 17.10

is cancellative, and hence If

a e U , then

implies that U .

U

e

is an identity

e e rad(e + u) =

e e a + U , and hence

Therefore

U

T

is in­

is a group.

To conclude this section, we show asymptotic torsion, then

a

that if

T is

free of

can be imbedded in a semigroup

that is a semilattice of subgroups. THEOREM 17.12. torsion and let

Assume that

T = u ,tT ueU u

T

is free of asymptotic

be the representation of F

a semilattice of its Archimedean components. quotient group of

Ty

for each

imbedded ^in a semigroup lattice of subgroups Proof.

such that

Wu , where

.

a,b e T y

a + d = b + c .

Then

Gu

Wu ^ Gu

for each Gu

[a,b] = [c,d]

Since the sets

Ty

be the

a semi­ u e U .

in the form

that is, as equivalence classes

and where

as

T can be

W = uu£yWu is

We represent the elements of

used in Section 1 — where

W

u e U

Let

T

[a,b]

,

if and only if

are disjoint, the groups

238 Gu

are also disjoint.

operation [c,d] of

+

on

W

e Gv , then

Gu+y .

[a'jb1]

Let

W = UU 6JJGU •

as follows. [a,b] + [c,d]

If

We define an e Gu

[a,b]

is the element

This operation is well-defined,

and

[c,d] = [c',d!] , then

c + d f = c1 + d

so

for if

on

and

[a + c,b + d]

W .

T . The operation on

Gufor each Since

W , it

G

u

u

c G v - u+v

and

= [a’+ c f,b! +

+ d ’] .

is associ­

agrees with the operation each

Gu

is a subgroup of

by the very definition of J

a semilattice of subgroups.

+

J

follows that W = uue^Gu

isa representation

of

on W

as

Finally, we show that the

natural inclusion mapping of identified with

W

e U , and hence

+ G

[a,b] =

a + b* = a 1 + b

Associativity of the operation follows because on

[a + c , b + d ]

that (a + c) + (b' + d ’) =

(a* + c ’) + (b + d)

ative

and

T

into

W , where

a e Tu

is

[2a,a] e Gu , is a semigroup homomorphism.

This is true since [2a,a] + Hence, we can take

[2b,b] = [2(a +

Wu = Gu for each

b),a + b]

u e U in the

. state­

ment of Theorem 17.12. Section 17 Remarks The problem of determining conditions under which a semigroup ring

T[U]

cases other than where U

is regular has been investigated in T

is a commutative unitary ring and

is a commutative monoid.

For example, Gilmer and Teply

show in [61] that the statement of Theorem 17.4 remains valid if the hypothesis that

R

is unitary is dropped;

reason we have chosen to state condition terms of Connell

p

being regular on

R .

If

(3) U

this is one

of (17.4) in

is a group, then

[36], using previous work of several authors, shows

239 that

T[U]

is regular if and only if

T

is regular,

locally finite, and the order of each element of unit of

T ; here neither

mutative.

T

nor

Finally, Weissglass

U

tive and

U

T

nor

is

is a

is required to be com­

[131] has considered the

problem of determining conditions under which regular, where neither

U

U

U

T[U]

is

is required to be commuta­

is not necessarily a monoid.

While Boolean rings form a subclass of the class of regular rings, the problem of determining the semigroup rings T[U]

that are Boolean can be solved using only first prin­

ciples. T

The result is that

is Boolean and

U

T[U]

is Boolean if and only if

is a semilattice.

conditions follows from the equality

Necessity of these

(tXu )2 = t2X 2u

fact that a Boolean ring is commutative;

and the

sufficiency follows

easily from the given conditions and the observation that T [U]

has characteristic

2

if

T

is Boolean.

The material in Section 17 concerning the representation of a commutative semigroup as a semilattice of its Archimedean components has been applied in work on questions concerning semigroup rings; [128-129], and [30].

see, for example,

[132],

240 §18.

Monoid Rings as Priifer Rings

In Section 13, conditions on a monoid domain

D[S]

have

been given in order that it should be a Priifer domain, a Bezout domain, a Dedekind domain, or a

PID .

In this section

we consider certain analogues of these classes of domains for rings with zero divisors. named are, respectively,

The analogues of the classes just the classes of Priifer rings, Bezout

rings, general ZPI— rings, and principal ideal rings

(PIR’s) .

At the same time, we consider here the classes of arithmetical rings and multiplication rings; an arithemetical integral d o ­ main is the same as a Priifer domain and the concepts of

multi­

plication domain and Dedekind domain agree, but these equivalences do not carry over to the case of rings.

Each of

the classes named is treated to some degree in [51], except for the class of Priifer rings.

On the other hand, the con­

cepts may be unfamiliar to some readers, so we develop in this section enough of the theory to facilitate our treatment of the monoid ring characterization problems.

More detailed

references are given in the Section 18 Remarks. Throughout the section, S

R

denotes a unitary ring and

denotes a torsion— free cancellative monoid.

We say that

R

is a Priifer ring if each finitely generated regular ideal of R

is invertible.

the property that

An invertible ideal An = (a^,...,a£)

it is clear that a Priifer ring

R

A = (a^,...,a^)

for each

n e Z+ .

has Thus,

satisfies the following

condition. (18.1) regular, then

If

a,b e R 2

and if at least one of 2

ab e (a ,b ) .

a

or

b

is

241 For the sake of brevity, we use the this section for a unitary ring (18.1).

ad hoc term

P— ring in

R satisfying condition

Theorem 18.9 shows that the monoid

ring

R [S ]

is a

Prufer ring if and only if it is a

P— ring. Theorem 18.2

lists three elementary properties of

P— rings.

THEOREM 18.2. system in

Let

(2)

If each

R^ is a

(3)

If R[S]

is

ae £

i e

Z+

Choose

a,b

P— ring, then e RN b 2

a R

(a ,b )

P— ring. is a

P— ring.

a

P-ring.

R is

with

3

regular, then there

is regular, 2

e R with b

such that a,b

R^ .

ab e

Thus

aR^ = aR^ , and

implies

that

€R^ , and 2

regular. b

is

There exists

of necessity regular

2

ab e (a ,b }R^ , which implies that

(a2 ,b2) . (3):

in

RN is

P— ring, then

such that

Then ab e

*

(a2 ,b2)RN = o 2 ,e2) . (2):

in

a

oo

R =

P— ring, then

(1): If a,3

a,b e R

= bRN .

a

an ascending se-

such that

If R

Proof.

is

R

(1)

exist

be a regular multiplicative

R , and assume that

quence of subrings of

3R n

N

R[S]

If

a,b € R

, and hence

with

is a

S

and that

G*

P— ring, then Proof.

R

.

b

is regular

But Theorem 12.2

is contracted from

R[S]

.

In

ab e (a2 ,b2) = ( a 2 ,b2}R[S] n R .

COROLLARY 18.3. of

regular, then

ab e ( a 2 ,b2}R[S]

shows that each ideal of particular,

b

Assume that

G

is the quotient group

is the divisible hull of R[G]

and

R[G*]

are

G .

If

R [S ]

P— rings.

It is immediate from Theorem 18.2 that

R[G]

is

242 a

P— ring.

G* {G

Moreover, the proof of Theorem 13.4 shows that

can be expressed as the union of an ascending sequence i m m =1

each

of subgroups, where °

R [Gm ]

R[G*]

is a

G m

r

is a

= G

for each

m .

Hence

P— ring, so Theorem 18.2 implies that

P— ring.

The next result is a generalization of Theorem 13.3. THEOREM 18.4. then either

R

Proof. such that S

is

If

Assume that

s

S

is not invertible. Xs

not invertible in follows that

S

is idempotent.

of

S

is a

P-ring,

is a

group.

r e R\{0}

Choose

is a regular element of rXs = r2f + X 2sg .

and since

s ^ Supp(X2sg) f^

R[S]

is not a group and choose

rXs e (r2 ,X2s) , say

coefficient

and if

a regular ring or

is cancellative,

Thus

S * {0}

S

.

R

Since

R [S ] . Since

r =

s

is

it

f°r some 2 (r) = (r )

f , and this implies that

We conclude that

.

is cancellative,

Therefore

s e S

is regular if

S

is not

a group. We subsequently show that R

R[S]

is a regular ring in any case.

a treatment of the

a P— ring implies that

The key to

case of the rational

THEOREM 18.5.

If

R[Q]

is a

r e R\(0>

.

this result is

group

P-ring, then

ring R

R[Q] is a

regular ring. Proof.

Choose

1 -X

is regular in

Since

R[Q]

{R[Z/n!

]

R[Q]

, so

Corollary 8.6 shows that r(l - X) e (r2 ,(l - X ) 2) .

is the union of the ascending sequence of subrings,

it follows that

.

243 r (1 - X) e ( r 2 ,(l - X) 2}R[X±1 /m ]

for somem e

change of variable we can assume r(l - Xm ) e (r2 , (1 - X m )2) ideal R[X]

in

A = (r2 ,(l - X m )2}R[X] , and hence

{X1} .

A

Therefore

r(l - Xm ) e A .

Z+ .

By

a

that R t X ^ " 1 ] = R[X]{xi}

is comaximal with

.

The

XR[X]

in

is prime to the multiplicative system A

is contracted from

We next observe

, so

that R [X ]

is

a free

R[Xm ]-module with a free basis consisting of(l,X,...,Xm Thus the ideal R[X]

B = ( r 2 ,(l - X m )2}R[Xm ]

by Theorem 12.2, so

R— automorphism of

R[Xm ]

this mapping we obtain

is contracted from

r(l - Xm ) € B . mapping

*}.

1 - Xm

There exists an to

rXm e { r 2 ,X2m}R[Xm ] .

Xm , and under An easy com-

2

putation then shows that

r € r R , and thus

R is regular,

as asserted. The results up to this point enable us to give equivalent conditions for the group ring H

to be a

R [H ]

P-ring or a Prtifer ring

of a torsion— free group (Theorem 18.8).

But in

order to broaden the statement of that result, we consider briefly the notion definition:

R

(A

n

n

B)

+

(A

is C)

of an arithmetical ring. We recall arithmetical if for all ideals

A n (B + C) = A,B,C

dition is equivalent to distributivity of lattice of ideals of

the

of R +

;

this

over

n

con­ in the

R , and it is also equivalent to the

validity of the Chinese Remainder Theorem in

R .

It is well

known that a unitary integral domain is arithmetical if and only if it is a Prufer domain.

We call a unitary ring

chained ring (or valuation ring) if the set of ideals of is linearly ordered under inclusion.

R

a R

244 THEOREM 18.6. if

R^

The ring

R

is arithmetical if and only

is a chained ring for each maximal ideal

M

of

R .

A Bezout ring is arithmetical and an arithmetical ring is a Priifer ring. Proof. ideal of

R., M

ideals of section,

Assume that R

R

is arithmetical.

is extended from to

Rx.

R

Since each

and sinceextension of

distributes over both sum and inter-

it follows that

R^

is also arithmetical.

Thus,

we assume without loss of generality that

R

is quasi— local

with maximal ideal

R

is a chained

M , and we prove that

ring.

It suffices to prove that the set of principal ideals

of

is linearly ordered under inclusion.

R

a,b e R

and

assume that b J

(a)

(a) = (a) n (b,a - b) = [(a) n Thus, write

a = u +v , where

v = ra = s(a - b) e (a) n then

a = c "''db e (b)

sb = (s - r)a 1 —c - s

with

.

we conclude that

and let

e^

ideals of chained, n .

.

to

and

If

R ,

c

is a unit of

c e M .

R , and since and that

(a - b ) ] .

ca = db e (a) n (b)

Moreover,

s e M .

implies that

t*le set and

R

u =

Otherwise,

a e (b)

converse, let

We have

(b)] + [(a) n

(a - b)

b { (a)

is a unit of

.

Thus, take

(1 - c - s)a = — sbe(b), R

is chained.

m ax im a l ideals

For

the

of

R

c^

denote extension and contraction of

R xt

for each

Mx

A e A

.

Since

RXjI

its lattice of ideals is distributive under

Thus,

Hence

for ideals

A,B,C

of

R

we have

is +

and

245 A n (B + C) = n, . [A n (B + C)]eXCA = A €A

n [Ae^ n (Be^ + Ce*)]c*= n,[(Ae* n Be*) + (Ae* n C6*)]0* = A

A

n. [(A n B) + (A n C)]eXcA = (A n B) + (A n C)

.

A

This completes the proof that if each localization Assume that

R

R^

Bezout ring T

T

is a chained ring.

M

of

is chained.

Let

elements

ar + b s , a = cx, and

x ^ M

and

is Bezout

or

R

be the maximal ideal of

a,b e T .

. Then

Since

If

(c) = (a,b) , c

If

=

c = crx + csy ,

c * 0 , 1 - rx - sy

y | M .

(b) c (a) .

M

r,s,x,y e T such that

b = cy

c(l - rx - sy) = 0 .

c € (a)

R^

R , and hence to prove that

and choose nonzero elements

T , so

Then

it suffices to prove that a quasi— local

then there exist

of

is arithmetical if and only

is a Bezout ring.

for each maximal ideal is arithmetical,

R

so

is a nonunit

x $ M , for example, then

Thus, a Bezout ring is arithmet­

ical . Finally, assume that

R

is arithmetical and let

a finitely generated regular ideal element

b e A ,

and let

of R .

first paragraph of this proof.

A

Since

R^ ^

is principal for each A 6 ^ [( b ) :A 6 ^] .

A e A ,

Because

(b)e A:AeA = [(b):A]e A .

A

and hence

be

Choose a regular

(M } , e , and c

A

A

A

be as

in

is chained, e (b) A =

the A6^

is finitely generated,

Therefore

(b) = nx (b)e XcA = nx (AeA[(b):A]e A)cA = nx (A[(b):A])eAcA = A[(b):A]

Since

(b)

is invertible, this proves that

.

A

is invertible,

2 46 and hence

R

is a Priifer ring.

THEOREM 18.7. nomial ring

If

Proof.If

that

(f,g)

f

R[X] and

for each

g

n , it also follows

It suffices

are nonzero elements of We assume that

deg f .

If

deg f = 0

then we can assume that

f

—

We assume that

(f,g)

be

the leading coefficient of

idempotent

generator of the ideal tR

f

thesis.

et

eg = q*ef + r d e g r < deg ef

of for

.

principal.

and

= n + 1 .

and let R .

e

be an

Then

R =

, and

(f,g) =

The ideal

ef

eR[X]

, the leading

is a unit since

someq,r e eR[XJ

It follows that

et« eR = eR . Hence

with

r = 0

(ef,eg) = (ef,r)

r * 0 , the induction hypothesis implies that principal, and if

Then

is principal by the induction hypo­

And as an element of

coefficient

f * fh

deg f

of

eR © (1 - e)R , R[X] = eR[X] © (1 - e)R[Xj

((1 - e)f , (1 - e)g)

and we

is principal if

deg f ^ n , and we consider the case where

.

, then

is idempotent.

g = (1 - f + fg)h .

(ef,eg) © ((1 - e)f , (1 - e)g)

R[X]

that i s , if

h = f + (1 - f)g , for

t

to prove

deg f £ deg g

(f,g) = (h) , where

Let

R [Q ] =

Hence, we prove that the poly­

is a Bezout ring.

use induction on —

Bezout ring,then its quotient

R[Z/n!] - R[Z]

is principal.

f e R

, and the

are Bezout rings.

is a Bezout ring.

nomial ring that if

is a

R[Z]

is also a Bezout ring. Moreover, since with

R[Q]

R[Q]

R[X]

R [Z ]

u^R[Z/n!]

is a regular ring, then the poly­

R [ X ] , the integral group ring

rational group ring

ring

R

r = 0 , it is clear that

or .

If

(ef,eg) (ef,eg)

Since each of the summands of the ideal

is is

(f,g)

247 in the decomposition (f,g) = (ef,eg) © (fl - e ) f , (1 -e ) g ) is principal,

(f,g)

is also principal.

This completes the

proof. THEOREM 18.8.

Assume that

group with divisible hull

H* .

H

is a nonzero torsion— free

The following conditions are

equivalent. Cl)

R[H] is a

C2)

R[H*]

C3)

R

is

P— ring.

is a

P— ring.

a regular ring and H*

(4)

R[H] is a

C5)

R[H] is arithmetical.

C6)

R[H] is a

Proof.

Bezout ring.

Prufer ring.

Corollary 18.3 shows that

Theorem 18.7 shows that cations 18.6.

(4) -**> (5)

(3)

and

implies

(5) = >

(2)

sum of copies of

implies

(3) .

Cl)

implies

(2) ,

(4) , and the impli­

C6)

Since each Prufer ring is a

prove that

=Q .

follow from Theorem

P— ring, we need only

The group

H*

is a direct

Q , and in view of Theorem 18.5, it suffices

to show that the assumption that there is more than one Q— summand leads to a contradiction. H*

=

of

Theorem 18.2 implies that

P— ring.

Q ©

Q © K , where

Hence

R[Q]

K

is a subgroup R[Q © Q]

plies

(3) .

* R[Q][Q]

ofH*

.

Part

is

a

is a regular ring by Theorem 18.5, and

this contradicts Theorem 17.4, for semigroup.

Thus, assume that

We conclude that

Q

is not periodic as a

H* = Q , and hence

(2)

im­

(3)

248 We are able to extend Theorem 18.8 to monoid rings with­ out further ado. THEOREM 18.9. that

S

Assume that

R

is a unitary ring and

is a nonzero torsion— free cancellative monoid.

The

following conditions are equivalent. (1)

R [s ]

is a Bezout ring.

(2)

R [S ]

is arithmetical.

(3)

R[S]

is a Priifer ring.

(4)

R[S]

is a

(5)

R

P— ring.

is a regular ring, and to within isomorphism,

is either a subgroup Proof. prove that

of

In view (4)

assume that

Q

or a Priifer submonoid

(5)

R[S] is a

and

(5)

S.

Hence

R

is regular and

sider

S

as a submonoid of

H* s Q Q .

of

Qq

containing

that

If

assume that

S

is a

P— ring.

We con­

is a subgroup of S

Q

is isomorphic to a

1 ; without loss of gene­

S = U .To show that

h * 0so that

(X2a ,X^k)

.

R[S]

a > b .

S is

a

Prtifer

a + b , we have

where

Since

X

a

a,b e S .

We

b and X

are

, a P-ring, we have

This implies that

a + b e 2b + S .

than

. Thus,

contains both

Take h = a - b e H n Q Q ,

regular elements of

or

only

bethe quotient R[H]

S

.

it suffices, by Theorem 13.5, to show that

H n Qq c s .

Xa+b e

(1)

by Theorem 18.8.

In the contrary case,

rality we assume monoid,

H

Theorem 18.2 implies that

positive and negative rationals, then

U

implies

P— ring and let

group of

submonoid

Q

of Theorems 18.6 and 18.9, we need

implies

by Theorem 2.9.

of

S

As each a+ b e

either

element of 2b + Sand

a + b

2a + S h =

e 2a + S

isgreater a — b

e S .

249 Thus

H n Q q c s , and (4)implies (5) = >

(1):

If R

, as asserted.

is regular and

Q , then Theorem 18.8shows the other hand, if

(5)

that

R[S]

S

is a subgroup

of

is a Bezout ring.

S is a Prufer submonoid of

On

Q , then

oo

write

S = u

S i “ ^i+1 R[S^]

, where each

^or each

* •

is isomorphic to

Therefore

R[S]

is a cyclic monoid and

Then R[X]

R[S] = u^RCS^]

, where each

, a Bezout ring by Theorem 18.7

is also a Bezout ring, and this completes

the proof of Theroem 18.9. The ring-theoretic analogue of a Dedekind domain is the notion of a general

ZPI— ring, defined as a ring in which

each ideal is a finite product of prime ideals. of

[51]

shows that a unitary ring

ZPI— ring if and only if

R

R

Theorem 39.2

is a general

is a finite direct sum of Dedekind

domains and special primary rings, where a special primary ring is, by definition, a local ideal.

PIR

with nilpotent maximal

(The term special principal ideal ring

sometimes used instead of special primary ring.)

(SPIR)

is

For our

purposes, the equivalence in terms of Dedekind domains and SPIR's

could be taken as the definition of a unitary general

ZPI— ring.

Note in particular that a unitary general

is Noetherian.

An ideal

multiplication ideal if that it contains —

A A

of a ring

T

is said to be a

is a factor of each ideal of

that is, if {AB^lB^

is the set of ideals of

T

is an ideal of

contained in

a multiplication ring if each ideal of ideal.

T

ZPI— ring

A ; the ring T

T} T

is

is a multiplication

It is clear that a regular ideal of a unitary ring is

a multiplication ideal if and only if it is invertible, and hence a unitary integral domain

is a multiplication ring if

250 and only if it is a Dedekind domain.

Moreover, each unitary

multiplication ring is a Priifer ring.

On the other hand, a

regular ring is a multiplication ring (for A g B in a ? regular ring implies A = A = AB ) , and hence a multipli­ cation ring with zero divisors need not be Noetherian. Theorem 18.10 determines necessary and sufficient conditions in order that general

R[S]

should be a multiplication ring or a

ZPI— ring.

While Theorem 18.10 is a summary result,

we do not repeat our standing hypotheses concerning S

and

in its statement. THEROEM 18.10.

The following conditions are equivalent.

(1)

R [S] is a

PIR .

(2)

R[S] is a general

(3)

R[S] is a multiplication ring.

(4)

R

Proof. PID's

ZPI— ring.

is a finite direct sum of fields and

isomorphic to

of

R

Z

(1) = >

and

(2) = >

or to

(3):

is

ZQ .

(2) :

SPIR's

S

Each

, and

PIR

is a finite direct sum

hence is ageneral

Dedekind domains and

ZPI— ring.

SPIR's

are multi­

plication rings, and a finite direct sum of unitary multipli­ cation rings is again a multiplication ring. general

ZPI— ring is a multiplication ring.

(3) “ > (4): ring.

Therefore each

Then

R[S]

regular ring and submonoid of

Assume that

Q .

is a multiplication

is a Priifer ring, and hence S

is either a subgroup of

We show that

that each proper prime ideal By Corollary 8 .6 ,

R[S]

1 - X

P

R of

R Q

is a or a Priifer

is Noetherian by showing R

is finitely generated.

is not a zero divisor in

R[S]

, so

251 CP,1 - X ) Since

P

is invertible, and hence finitely generated. is the image of

(P,l - X )

tion map, it follows that

P

under the

augmenta­

is also finitely generated.

A

Noetherian regular ring is a finite direct sum of fields — say

R = F 1@...©Fn .

F^[S]

Thus

R[S] = F ^ [S ]©...©Fn [S ] , and each

is an integral domain that is a multiplication ring,

hence a Dedekind domain. that either

S = Z

(4) = > fields

(5):

F^

S = Zq .

If

and if

PID , and hence

or

Applying Theorem 13.8, we conclude

R = F 1©...©Fk

S

R[S]

is Z

or

is a direct

sum of

Z q , then each F^[S]

= £* 0 F ± [S]

is a

is a

PIR .

Section 18 Remarks Some detailed references on topics treated in Section 18 are the following. rings

[45],

tion rings

Prufer rings:

[82]; general [106],

[108],

[27],

ZPI-rings [56],

[65];

[107],

[66], [4].

arithmetical

[13]; multiplica­ Moreover,

[91,

Chapters IX and XJ is a general reference for these topics. We remark that a is a ring

T

ZPI— ring, as opposed to a general

in which each nonzero ideal is uniquely e x ­

pressible as a finite product of prime ideals unitary, then factors of uniqueness).

ZPI— ring,

T

A unitary ring

(if

T

is

are disregarded in considering T

is a

if it is either a Dedekind domain or a

ZPI— ring if and only SPIR .

The choice

of terminology comes from the German Zerlegung Primideale. Much of the material in Section 18 stems from

[59], but

new proofs have been necessary for most of the results of that paper.

This is because Gilmer and Parker in [59] made

strong use of a result of Griffin in [65] stating that a

252 unitary ring

R

is a Priifer ring if and only if

integrally closed

P— ring.

R

is an

While it is true that a Priifer

ring is an integrally closed

P— ring, the status of the con­

verse is in question; Griffin's proof shows only that in an integrally closed

P— ring

(a^,...,an ) , with

R , each ideal of the form

ai > ,,,»an -i

regular, is invertible.

P— ring need not be integrally closed, and Priifer ring.

For example, if

elements and if domain closed.

F

D = F + X K [[X ]]

is a

hence need not be

Kis the Galois

is the prime

A

subfield of

field with K, then the

P— ring, but is not integrally

For details of this example, as well as related r e ­

sults, see [112],

[27], and [53].

a 4

253 §19. Where

S

Monoid Rings as Arithmetical Rings —

the Case

is not Torsion— Free

Results of Section 18 show that if

R

is unitary and

is torsion— free and cancellative, the monoid ring Priifer ring ring.

iff

it is arithmetical

Moreover,

R [S ]is a general

multiplication ring

iff

it is a

iff

is a

it is a Bezout

ZPI— ring

PIR .

R[S]

S

iff

it is a

An examination of

the proofs in Section 18 suggests that the assumption that is torsion— free is used sparingly,

and that it may be possible

to extend some of the results to the case where cancellative monoid.

S

S

is any

This section is devoted to that end.

The key concept in this development turns out to be that of an arithmetical ring, rather than that of a Priifer ring.

Our

results show that even for finite groups, a group ring that is arithmetical need not be a Bezout ring, and that general a

ZPI— ring does not, in general, imply that

R[G] R[G]

a is

PIR . The first four results of the section are devoted to a

determination of conditions under which a monoid ring is quasi-local

(that is, has a unique maximal ideal) or a chained

ring; these results do not require that the monoid in question is cancellative. mining conditions ways: R[S]

They are related to the problem of deter­ under which

first, a chained

R[S] is arithmetical in two

ring is arithmetical, and second,

is arithmetical if and only if each localization

(R[S])m

of

R[S]

at a maximal ideal

M

is chained.

In

connection with the latter condition, we remark that for a multiplicative system R^[S]

N

in

R , the rings

R[S]^

and

are canonically isomorphic for any unitary ring

R

and

254 any monoid

S .

Throughout the section,

R

denotes a unitary ring and

S

denotes a cancellative monoid. THEOREM 19.1. monoid ring

R[U]

is a

maximal ideal

M

a unit of

F[U]

is a group.

and

and

Let

H

u

particular,

F[U]

U

Then

implies that

Let

Therefore

R[U]

R

is

Xu

is

Therefore

U

F [U ] ; in

is quasi-local. it follows that

p = char(F)

Thus, choose

b = 1

has characteristic

, where

, so

be a prime that divides

X^ - 1 = b ( X - l ) ^

is monic,

Xu

is contracted from a

F[H]

p— group.

To prove the converse, of

p

1-

Theorem 12.2

let I

the ;

u € U

X? — 1

for some unit

and

this of

is quasi-local.

is the only prime factor of

X^ -1 F

F[H]

Then

Then

is contracted from

F[(u)] ^ F [ X ] / ( X P — 1)

X-l

U .

U .

U . We show that

is a

F [X ] , and therefore Since

u
is not quasi-local,

order of an element of

F = R/M .

(1 — Xu ) + (Xu ) = F[U]

F[H]

Since

is also quasi-local.

and let

, and hence

is a torsion group.

p .

R

be a subgroup of

F[Z] - F[X,X *]

will imply that

,

R

each maximal ideal of

maximal ideal of

F .

R[U]

is invertible in

shows that each ideal of

order

is

is quasi-local.

is quasi-local. Pick

is a nonunit ofF[U]

U

R[U]

be the maximal ideal of

F [U ] = R[U]/M[U]

Since

if and only if R

The

M , char(R/M) = p * 0 , and

Assume first that

is a homomorphic image of

Let

is a nonzero monoid.

p— group.

Proof. R

U

is quasi-local

quasi-local with U

Assume that

This in

b

of

X^ - 1 = (X - l )*5 .

p . be the

augmentation

quasi-local with maximal ideal

ideal M ,

255 char(R/M) = p R[U]/(M[U] Let

P

+ I)

P n

so

and each

p— group.

Since R .

P = M[U] + I .

Then

is maximal in R[U]

R[U]

.

is integral over

Therefore

P

n R=

M

is generated by

1 - Xu

has characteristic

is nilpotent modulo p .

Consequently,

M[U] I E P ,

This completes the proof of Theorem 19.1.

COROLLARY 19.2. R[U]

.

. The ideal I

{ 1 — Xu | u e U) R/M

R[U]

R is maximal in

P p M[U]

since

U is a

=; R/M , so M[U] + I

be maximal in

R , then and

* 0 , and

If

U * {0)

, then the monoid ring

is a local ring if and only if

residue field of characteristic

R

p * 0

is a local ring with and

U

is a finite

p-group. Proof.

The result follows immediately from Theorem 19.1

and the fact that

R[U]

Noetherian and

is finitely generated (Theorem 7.7).

U

is Noetherian if and only if

In order to determine conditions under which

R

R[U]

is

is a

chained ring, we use an auxiliary result from group theory. For a prime

p , the

p-quasicyclic group is the multipli­

cative group of complex n e Z+ .

If

G

pnth

roots of unity, taken over all

is quasicyclic or cyclic of prime-power

order, then it is well known that the set of subgroups of is linearly ordered under inclusion.

G

Corollary 19.3 estab­

lishes the converse. THEOREM 19.3.

If the set of subgroups of the group

is linearly ordered under inclusion, then

G

G

is either cyclic

of prime-power order or a quasicyclic group. Proof.

To obtain the result, we need not assume that

G

256 is abelian in advance;

in fact,

a^,...,an e G , then the

subgroup of

G

groups

.

Since the subgroups of

is

a torsion group.

Ca^)

ordered,

G

generated by

if

(a^

is one of the cyclic Zare

Hence

G

not linearly is the direct sum

of its primary components.

It is clear, however, that

indecomposable,

so that

is a

Let

subgroup of

(x^)

be a

we're finished.

G

order

p

.

p— group for some prime of order p .

Otherwise, choose

E Cy2) > and i n fact, k+1

G

G p

.

If G = (x^)

,

y 2 e G\(x-^)

.

Then

= (p y 2) , where

(x^

It follows that there exists

is

y2

has

x 2 e (y2)

of

2 order

p

such that

complete, and if

x^ = p x 2 .

G = (x 2^ * ^ e

If

P ro°f is

G * (x 2^ > we can continue the process.

Hence,

if

Gis not cyclic,

then there exists a

sequence

{Xi>“

in

G

has order

Px ^+^ = x^

for each G

such that x^

i > 1 .

contains

tained in

Let

H = u~(x^)

p1

.

and

No cyclic subgroup of

H , and hence each cyclic subgroup of H .

Therefore

G = H , and

G

is the

G

is con­

p-quasi-

cyclic group in this case. THEOREM 19.4. monoid ring field

R[U]

Assume that

either cyclic of order

pn

Assume that

p *

0 ,

or R[U]

and that

p— group, where

that ideals

R

is a field. (m)

and

is

R

is a

a group,

is a chained ring.

R

is a

U

if

Thus, pick of

1 - Xu ^ (m) , it follows that

and

Theorem

is quasi-local with maximal ideal

( 1 - X U)

The

p—quasicyclic.

19.1 implies that U

is a nonzero monoid.

is a chained ring if and only

of characteristic

Proof.

U

p = char(R/M) m e M

R[U]

and

.

m e (1 - X u ) .

We show

u e U\(0)

are comparable.

M

. The

Since

Under the

U

i

257 augmentation map,

(1 - X u )

to itself.

m

Hence

characteristic

goes to

(0)

and

and

R

is a field of

Since the set of kernel

ideals of R[U]

= 0 , M = (0) ,

p .

m

is linearly ordered, the set of congruences on and hence the set of

subgroups of U

Theorem 19.3, we conclude that pn

or

U

is mapped

U

is chained,

forms a chain.

is either

By

cyclic of order

p—quasicyclic.

If

R

is a field of characteristic

p— quasicyclic, then

U =

subgroup of

U

f,g e R[Un ]

for some

, where

of order

p1 .

n .

If

p

and

IK

pn .

is

is the cyclic

f,g e R[U]

, then

Hence, to prove that

chained, it suffices to consider the case where cyclic of order

U

R[U]

is

U = Un

is

In that case,

R [U] = R[X]/(Xp n -l) = R [X] / (X - l)pn is a

SPIR

Dn (X-1)/(X-1)H , hence a

with maximal ideal

chained ring.

This completes the proof.

We turn to the problem of determining conditions under which

R[S]

is arithmetical

be cancellative).

(recall that

S

is assumed to

Theorem 19.5 lists some basic properties

of arithmetical rings. THEOREM 19.5.

Let

N

and assume that R

such that

be a multiplicative system in

a directed family of subrings of

R = u. TR. . 1€ I 1

(1)

If

R

is arithmetical, then so is

(2)

If

R

is arithmetical,

If

each

then

R

is

RN . integrally

closed. (3)

R ,

R^

is arithmetical, then

R

is

258 arithmetical. (4)

Assume that

T

is an extension

free

R—module basis containing

then

R

1 .

ring of

If

T

R

with a

is arithmetical,

is also arithmetical.

Proof.

(1) follows from the facts that

R^

is extended from

to

R^

R

and that extension

distributes over both

Since

R

+

of ideals

of

R

n .

and

is a Prufer ring if

each ideal of

R

is arithmetical, it

suffices to prove the statement in the Section 18 Remarks to the effect that a Prufer ring an element

y

gral over

R .

R

of degree

If

ofR

invertible,

arithmetical,>

e R and

R

Then a,b e l [(a)R. i

e R^

Since

F = F

implies

is integrally closed. that R

[(a) n (b)] + [(a) n (a -b)]

a

F =

2

so

The proof of Theorem 18.6 shows

a,b e R .

Fix

that is inte­

is both regular and idempotent.

Hence y

metical if a e

Take

is a root of a monic polynomial over

ring, F is

F = R . (3)

y

R

n + 1 , then the fractional ideal

is a Prufer

that

is integrally closed.

in the total quotient ring of

(l,y,...,yn ) R

R

for some

n (b}R.l i J +

is

for all i

. n

[{a)R. i

L

arith­ a,b e R .

Since

R^

( a- b}R.] i J

f

is

, and

this implies that the corresponding relation also holds in (4): then is

If

AT = E ^ j A x ^

AT n BT = ( SAx^

take idealsA,B,C Ris contracted [An

R-module basis for

for each ideal

free, it follows that

then

T

is a free

if

n (E Bx ^

are ideals of from

A

A

of

and

B

R .

.

T ,

Because

(x^)

areideals ofR ,

= Z (A n B)xi = R

R .

Since

(A n B)T

.

Now

each ideal of

T , it suffices to prove that

(B + C)]T = [(A n B) + ( A n C)]T

.

Using

is arithmetical, this is straightforward:

the fact that

259 [A n (B + C)]T = AT n (BT = (A n B)T

H

Assume that

is a subgroup of

then so are

= (AT n BT) + ( AT n CT)

+ (A n C)T = [(A n B) + (A n C) ]T

COROLLARY 19.6. and that

+CT)

R [G ]

and

S

S .

R[H] .

has quotient group If

R[S]

G

is arithmetical,

In particular,

R

is arith­

metical . In giving conditions under which where

S

R [S ]

is arithmetical,

is not torsion— free, we separate the cases where

S

is, or is not, periodic. THEOREM 19.7. nor periodic. H

Assume that

Let

G

If

R[S]

arithmetical, (2)

R

is arithmetical,

Proof. shows that

(1):

If

R[S]

H £ S , R[S/H] R[H]

are satis­

H £ S .

Since

then (19.6)

Thus H

Moreover,

S

is inte­

is integral over R[S/H]

it is a homomorphic image of

R[S]

is arithmet­ .

Because

is torsion— free and cancellative, Theorem 18.9

shows that

R

is a regular ring.

To show that

R[H]

regular, we must show that the order of each element is a unit of

R .

generated by

{h,g}

19.5 shows

is

is regular.

(1)

R[S] isarithmetical,

is integrally closed.

S , it follows that since

and let

is a Bezout ring, hence arithmetical.

grally closed by Theorem 12.10.

S/H g G/H

then

is a regular ring, and

R[S]

S

G .

Conversely, if the conditions in

fied, then

ical

is neither torsion— free

be the quotient group of

be the torsion subgroup of (1)

S

that

Pick is

R[G]

g e G\H . (h) ©

(g)

The subgroup .

Part

(1)

is h e H

K

of

G

of Theorem

is arithmetical, and Corollary 19.6

260 implies that

R[K] = R[(h)][Z]

is also arithmetical.

ing either Theorem 18.8 or the part of we conclude that h

is a unit of To prove

R[(h)] R

(1)

already proved,

is regular, andhence the order of

(Theorem

17.4).

(2) , we show that

This

R[S]

ascending sequence of Bezout rings.

establishes

case,

S

Since

or a Prufer submonoid of

is a group and

sequence that

Q

S/H

{(sn + H ^ n = l

S = un=it(sn ^ +

S/H

(s r ) n H = {0} .

where

R[( s r ) © H] = R[H][Z]

This proves that

R[S]

Q .

S/H

is a

In the first

is the union of an ascending

Moreover,

since

.

is torsion— free

nonzero subgroups. *

(1)

is the union of an

and cancellative, Theorem 18.9 shows that either subgroup of

Apply­

Therefore

It follows

(sR ) + H = (sR ) © H

R[S] = u n=iR [(sn ) ®

is a Bezout

ring for each

is a Bezout ring if

S/H

* n

.

is a group.

The proof in the other case is similar; there S/H = OO 0 un = l<sn + H> is the union of an ascending sequence of cyclic monoids and

oo

S = un= 1 [<sn >

R[<sn >° © H] - R[H][Zq ] well, this establishes The conditions in

Q

© H]

.

Since

is a Bezout ring in this case as (2) . (1)

more explictly in terms of

of Theorem 19.7 could be stated R,S,

and

H , but we choose not to

do so, for Theorem 18.9 gives conditions on

S/H

that

regular, and

R[S/H]

should be arithmetical for

R

in order

Theorem 17.4 provides necessary and sufficient conditions on R

and

H

for

R[H]

to be regular.

A cancellative periodic monoid is a group, and hence the case remaining from Theorem 19.7 in the problem of determining when

R[S]

is arithmetical is that in which

S

is a torsion

261 group.

Theorem 19.9 provides a reduction in this case.

THEOREM 19.9. of

Let

t*ie set

R , and assume that

S

is a torsion group.

is arithmetical if and only if each Proof. R[S] r \m

If

R[S]

~ Rj^j [S] A

A converse,

let

M

say

. It follows that

= M^

localization of

R.. [S] Mx

A .

R[S]

R [S]M

.

For the Since

R[S]

is maximal in

R [S ]M

R

—

is isomorphic to a

at a maximal ideal.

is arithmetical,

RLS]

then

be a maximal ideal of M n R

Then

is arithmetical.

is arithmetical for each

R , the ideal

R^[S]

R^ [S] MA

is arithmetical,

is integral over M n R

maximal ideals

Hence if each

is chained and

R[S]

is also

arithmetical. THEOREM 19.10. that p

, let

(1)

If

Moreover,

Sp

S^

be the

R[S]

is a torsion group and

for some

Foreach

such that

R

is arithmetical.

S^ * {0}

A , it follows that

S .

RXJf Mx

and

p *

is a field

is either cyclic or quasicyclic.

(2)

The converse of

Proof.

(1):

char (R/M.) A image of RXif [S ] .

and

R

is also valid.

is arithmetical.

S * {0} . p

S , the ring Theorem 19.1

quasi-local.

(1)

We have already observed that

arithmetical implies

p

p

R .

p—primary component of

is arithmetical, then

for each prime

char(R/M. ) X

Mx

S

is the family of maximal ideals of

prime

and

Assume that

Therefore

Since

R[ ]

S P

Assume

that

p =

is a homomorphic

is arithmetical, as is

shows, however, that R^ [S^]

R[S]

Rx,

[S ]

Mxl p J

is a chained ring, and

is

262 Theorem 19.4 shows that

Rw

is a field and that

S

is P

A either cyclic or quasicyclic. (2)

To prove that

ditions of

R

is

char(R/M) family

the con­

(1) , it suffices to show that each

arithmetical. that

R [S ] is arithemtical under

[S] is

Thus, we assume without loss of generality a chained ring with maximal ideal

.

The group

CT^^^€ j

Hence, part

S is the directed union

°f finite subgroups, and

(3)

M. of

Let

c =

its

R[S] = u^ejR[T^]

.

of Theorem 19.5 shows that it suffices to

prove that each

R[T^]

is arithmetical.

notation, we assume that

S

By a change of

is finite.

If

c divides the

order of S , then the hypothesis implies that

R

and

for some subgroup

T

Sc of

is cyclic. S, and

Moreover,

S = Sc © T

R[S] = R [Sc ] [T ] ,

where

R[Sc ]

ring with residue field of characteristic Hence, we also assume that S .

The group

S

c

ideals of istic

R[U^]

chained

by Theorem 19.4.

does not divide the order of

prime-power order.

R[U^]

c .

c

is a

can be expressed as a finite direct sum of

cyclic groups ideal of

is a field

lies over

M in

Each maximal

R , and hence all

maximal

have associated residue field of character­

By induction,

it suffices to prove the following

result, which we state separately for the sake of clarity. (19.11) if

If

R

is a chained ring with maximal ideal

c = char(R/M) , and if

power order

m = p

S

, where

is a finite group of prime c * p , then

R[S]

is arithmet­

ical .

Proof of (19.11). each maximal ideal of

Since R[S]

M ,

R[S]

is integral over

lies over

M

in

R ,

R , so the

263 maximal ideals of with those of Since

R[S]

are in one-to-one correspondence

R[S]/M[S]

* F[X]/(Xm -l)

c * p , the ring

F[X]/(Xm -l)

finitely many maximal ideals. maximal ideals of between R[S]

and

1

to

R[S]M

f e M[S]

R[S]

m

denote extension

•

Fix

of

ideals of

M[S]e = n?_-,M? = M e. . 1 1 1 j

.

Thus

is a coefficient of

Since

f .

Then

j

Take f

f = mf^

(f)e = (rafj)6 = (nOe .

that

, where

It follows

is the set of principal ideals of

R

is a chained ring, we conclude that the

set of principal ideals of R[S]j^

M[S] =

. We have

{(ra)e | m e M}

R[S]M . .

set

e

j

.

is reduced and has only

n and let

generates the content ideal of

that

F = R/M

Let

; we have

and assume that

fj £ R[S]\M[S]

, where

R[S]^

is chained, and hence

is also a chained ring. j

The results up to this point are sufficient to show that a group ring

R[G]

need not be arithmetical if it is a

Priifer ring and that

R [G ]

need not be a Bezout ring if it

is arithmetical.

For the first example, let

of characteristic

p * 0

not cyclic.

and let G

be

a

trivially a Priifer ring, but Theorem 19.4. D

R[G]

p— group that is

Hence

take

is a Dedekind domain that is not a p = 11

Let

The domains

R = D[l/p] R

group of order

.

arithmetical, but

is a is

D = Z[/-5]

R and

D

PID .

have Let

.

PID , and it is D .

the same class G

Theorem 19.10 shows that

R[G]

R[G]

generates a prime ideal of

also fails to be a 11 .

R[G]

is not arithmetical by

For the second example,

easy to show that

group, so

be a field

Theorem 19.1 and its proof show that

local ring with nilpotent maximal ideal.

Then

R

be the cyclic R [G ]

is not a Bezout ring since

is R

fails

264 to have this property.

Note that

R[G]

is, in fact, a

Noetherian arithmetical ring; the next result shows that such rings are general

ZPI— rings.

THEOREM 19.12.

The following conditions are equivalent.

(1)

R is a general

(2)

R is a Noetherian arithmetical ring.

Proof.

If

(1)

ZPI— ring.

holds, then

sum of Dedekind domains and

R = R1@...©Rn

SPIR's

.

is a direct

Each of the summands

is Noetherian and arithmetical, and it is straightforward to show that the direct sum has the same two properties. (2) a

=>

PIR ,

(1):

and hence is either a

valuation domain. to R

We note that a local arithmetical ringis

R ,

If

P< M , then

P R^

and

P—primary ideal, P

tained in

P

i .

primes Q. < Q. x i

M .

is contained in each

Let

(0) = n i= iQ^

(0)

Thus,

P.

i

9

R

It follows that R - E i © (R/Q^)

If

P^

in

P

is the only

M—primary ideal of R

properly con­

be a shortest primary r e ­

R , where

since a relation P^< P^

a maximal ideal of

local,

It follows that

is

P^-primary for

There are no inclusion relations among distinct P^

.

are proper prime ideals of

is the unique prime ideal of

presentation of each

M

is local and arithmetical, hence a

Noetherian valuation domain.

R , and

or a rank-one discrete

This observation has several applications

asfollows.

with

SPIR

and

P.

j

If

are comaximal for

i

* j since

J

properly contains at most one prime ideal. and

.

would imply that

P^

are also comaximal, and hence is

maximal in

R , then R/Q^

zero-dimensional, and arithmetical, hence a is not maximal, then P^

=

and

SPIR .

R/Q^

is a

is

265 Dedekind domain.

Therefore

R

is a general

ZPI— ring.

In view of Theorem 19.12, equivalent conditions for R[S]

to be a general

ZPI— ring can be easily obtained from

Theorems 19.7 and 19.10. THEOREM 19.13.

Assume that S

is not torsion— free. and let

H

Let

If

S = Z ©

order of

H

H

or

R

R .In

If

if

is a finite group, and Proof.

(1):

R

R[S]

is

Assume that

R[S]

a subgroup of

since

and

S , G,H ,and

that

R is

order of

R[S] S/H

and

rated, so

R[S]

S

= G

R[H]

ZPI—ring.

are regular, Q .

is Noetherian

are finitely generated.

It follows

fields,

R , and

S/H - Z Q

is Noetherian and is Noetherian.

Theorem 19.13 shows that

R [S ]

H is finite,

S/H - Z

Conversely, if R ,S ,and R

ZPI— ring,

R

S - H © Z , and similarly,

(1) , then

general

is a general

the proof of Theorem 19.7 shows that if

in

is a

is Noetherian, then

is a unit of

S - H © Zq .

a

Q or a Priifer submonoid of

a finite direct sum of

H

R [S ]

arithmetical.

and

Moreover,

R [S ]

is a general

shows that H c S , R

is

is finite and the

PIR .

Theorem 19.7 S/H

H

this case,

S is periodic, then

ZPI— ring if and only

is a general

direct sum of fields

S =:Z ^ © H , where

ZPI— ring implies that it is a (2)

R[S]

is a finite

is a unit of

S

G .

S is not periodic, then

ZPI— ring if and only if

S

be the quotient group of

be the torsion subgroup of

(1)

and

G

is nonzero andthat

or

S

Z Q . Finally,

S/H - Z , then

implies H

the

that

are as described is finitely gene­

Moreover, part

(2)

of

is a Bezout ring, from which

266 it follows that

R[S]

Statement

(2)

is a

PIR .

follows immediately from (19.10), (19.12),

and the fact that R[S] is Noetherian if and only if Noetherian and

S

R = Z[/— 5, 1/11], G = Z ^

R[G]

ZPI— ring.

need not be a

nonzero and not torsion— free?

shows that a

PIR if it is a general

Under what conditions is

R [S ]

Part

a

(1)

PIR , for

S

periodic.

We proceed to consider the case where

periodic.

If

R[S]

is a

PIR , then

is finitely generated, hence finite.

and each

that

R

R^[S]

is a

PIR

S

for each

where

(1)

R

a field, and

R[S]

is a field, (3)

R

is a

is

PIR

PID’s

and

and

if

is a finite i , then

S

is a

PIR

—

is a

SPIR's ,

R = group such

R[S]

Thus, there are three cases to consider for conditions under which

S

It follows that

is a finite direct sum of

is so decomposed and

is not

is a

R^[S] is a PIR . And conversely,

R^©...@Rn

S

of Theorem 19.13

gives equivalent conditions in the case where

R = R.6...0R 1 n

is

is finitely generated.

The example group ring

R

is a

R

PIR .

in determining

namely, the cases

(2)

R

SPIR

that is not

PID

that is not a field.

The

next three results treat these three cases. THEOREM 19.14. istic F[G]

pand that is

a

only if the Proof.

G

PIR .

is a field of character­

is a torsion group.

p—primary component If

F

If p * 0 , then

F[G] is a

19.10 implies that G

Assume that

Gp

F [G]

G^

of

If is a G

satisfies the given conditions,

p * 0 .

then

F[G]

= 0 PIR

, then if and

is cyclic.

PIR , then part

is cyclic if

p

(1)

of Theorem

Conversely, if is arithmetical

267 by Theorem 19.10, and finitely generated

F[G]

is Artinian since it is a

F-module.

Therefore

F[G]

direct sum of zero-dimensional local rings. summands is a local chained ring, hence a also a

19.15.

a field and that is a

PIR

Proof.

Assume that

G

PIR , so

if and only if

If

m

Theorem 19.14, is a

m

is a unit of

R[G]

PIR

R is a

F[G]

is

SPIR that is not

is a finite group of order

Theorem 19.10 shows that

R[G]

Each of these

PIR .

THEOREM

R[G]

is a finite

R[ G]

m .

is a unit of R , then part

Then

R . (2) of

is arithmetical.

As in

is also Artinian, and the proof that

is the same as that given in (19.14) for the

corresponding case. If

m

is not a

maximal ideal M m .

Since

of

Gp *

of Theorem

unit of R , then R

{0}and

THEOREM

p

R^ = R

a field, part

D[ G]

(2)

m

is

the exponent of

Proof.

(1)

arithmetical, and

Assume that

D is a

K .

Let

PID G

that is

dis­

be a finite group

The following conditions are equivalent.

(1)

unity over

is not

is a divisor of

PIR .

19.16.

m .

is not

R[G]

tinct from its quotient field of order

belongs to the

and char(R/M) =

19.10shows that

hence is not a

m

is a

PIR .

a unit of G

and

K , then

D ,and if

u

is a primitive

D[u]

is a

(1) = >

some prime divisor

of

m

is

is any divisor kth root

of

of

PID .

(2): If p

k

mis not a

unitof

the characteristic of

D ,then D/M A

268 for some maximal ideal

of

D .

Since

is not a A

field, part

(1)

of Theorem 19.10 shows that

arithmetical in this case. m

is a unit of

D .

If

Therefore k

D[G]

D[G] a

is not

PIR

implies

divides the exponent of

G ,

then the fundamental theorem of finite abelian groups implies that there exists a subgroup cyclic of order D[G]

, it is a

D[u]

k .

Since

PIR .

Thus

is isomorphic to

K

of

G

D[G/K]

such that

G/K

is

is a homomorphic image of

D[X]/(Xk -l)

is a

PIR .

D[X]/(f(X))

, where

minimal polynomial for

u

over

D

(which is equal to the

minimal polynomial for

u

over

K)

is a homomorphic image of

D[X]/(X

\r

f(X)

Since

is the

, it follows that

D[u]

— 1) , and is therefore a

PID . (2) = >

(1):

Since

m

is a unit of

Theorem 19.10 implies that D[G]

is a general

D [G ]is arithmetical.

ZPI— ring by part (2) of

over, Theorem 9.17 implies D[G]

is isomorphic to the

{D [

G

D[G]

.

D[G]/P^

]

, where

{

integral over P. n D = (0)

D

Hence

(19.13).

is reduced.

M or e­

Hence

^

is the set of minimal primes of is a

PIR , we show that each D

is integrally closed,

are regular in

D[G]

, and

D[G]

is

D ; hence the lying under theorem implies that for each

D[{Xg + P- | g e G } ] . cyclic, generated by

u

(1) of

direct sum of the rings

PID .The domain

nonzero elements of

D [G ]/P^

that D[G]

To show that D[G] is a

D , part

i .

Therefore

^

Consider first the case where h .

Then

as a ring extension of

is a primitive

D[G]/P^

u = X*1 + P^ D , and

G

generates

um = 1 .

Therefore

kth root of unity for some divisor

m , and the hypothesis of

(2)

implies that

is

D[u]

k

is a

of PID .

269 To summarize,

if

G

is cyclic, w e ’ve shown that

isomorphic to a finite direct sum each of

u^

is a root of unity in a fixed algebraic closure

In the general case, express

G

r > 1 .

Let

H = Gj®...@Gr _^ .

and the exponent of exponents of

H

G

u^

divides

G = G-j©...©Gr ,

Then

G = H © Gr ,

is the least common multiple of the

and

Gr .

By induction we assume that

is a finite direct sum of domains

D[v^]

, where

is a

direct sum of rings

thesis on

D

order of

Gr

D[v^][Gr ] , where

implies that each is a unit of

proof in the case where

D[v^]

is a

D , hence of

G

D[v^]

Wj e L

is

(= order) y—

e L

is

H . Thus the hypo ­

PID .

.

The

Thus, the

is cyclic implies that

is isomorphic to a finite direct of rings

D[H]

v^ e L

a root of unity whose order divides the exponent of D[G]

L

as a finite direct sum

of cyclic groups of prime-power order; say where

is

D[u^]@...©D[un ] , where

K , and where the (multiplicative) order of

m .

D[G]

D[v^,w^]

D[v^][Gr ] , where

a root of unity whose order divides the exponent of

Gr .

We have

D[v^,Wj]

= D[y^^] ,

where

is a root of unity whose order is the least common

multiple of the orders of the exponent of

G .

D[G] = E • -©D [y - -]

1>J

SPIR

and

Thus each

is a

1J

Since an

v^

w^. , hence a divisor of

D[y^j]

PIR .

is a

PID , and

This completes the proof.

is a local chained ring, Corollary 19.2

and Theorem 19.4 yield the following characterization of monoid rings that are THEOREM 19.17. ring

R[U]

is a

characteristic

SPIR's If

SPIR

p * 0

U

.

is a nonzero monoid, the monoid

if and only if and

U

R

is a cyclic

is a field of p— group.

270 Section 19 Remarks The question of whether each chained ring

R

is the

homomorphic image of a valuation domain has been raised in the literature.

If

R

is Noetherian, an affirmative answer

follows easily from Cohen's structure theorem for complete local rings.

Ohm and Vicknair show in [113] that the question

also has an affirmative answer if

R

is a monoid ring; their

proof is based on Theorem 19.4. A unitary ring

R

is said to be semi-quasi-local if

has only finitely many maximal ideals, and if

R

only if

R

is semilocal and

quasi-local if and only if

(2)

is semilocal

is Noetherian with only finitely many maximal ideals.

It can be shown that a group ring

G

R

R

G

R[G]

is semilocal if and

is finite;

(R;M^,...,M )

is a torsion group, and either

(1)

G

char(R/M^) =...= char(R/Mn ) = p * 0

R[G]

is semi-

is semi-quasi-local, is finite, or and

G/G^

is

finite. The problems of determining conditions under which

R[S]

is a Prufer ring or a Bezout ring seem to be open in the cases not covered in Section 18.

Glastad and Hopkins have

addressed the problem of determining when in [62].

R[S]

is a

Specifically, they consider the case where

PIR R[S]

a zero-dimensional ring of the form R fX j, ...,Xn ]/({X®i(1 -X? i) }" =1) , where each f^

et

and each

is a positive integer; each zero-dimensional monoid ring

that is a

PIR

type described.

is the homomorphic image of a ring of the

is

271 §20.

Chain Conditions in Monoid Rings

Two chain conditions —

the ascending chain condition

and the ascending chain condition for principal ideals — have already been considered in monoid rings, the latter for monoid domains. R[S]

To wit, Theorem 7.7 shows that a monoid ring

is Noetherian if and only if

R

is Noetherian and

S

is finitely generated, and Theorem 14.17 shows that a monoid domain

D[G]

and only if of

G

, where D

G

is a group, satisfies

satisfies

is of type

a.c.c.p.

(0,0,...

) .

a.c.c.p.

if

and each nonzero element

In this section we consider

certain other chain conditions in

R[S]

ically, the first part of the section

and

R[G]

.

Specif­

(through Theorem 20.8)

is concerned with the problem of determining conditions under which

R[S]

is Artinian or an

RM— ring.

The remainder of

the section treats the property of being locally Noetherian in unitary group rings. Throughout this section we use the notation unitary ring, ring

R

S

G

for a group.

is Artinian if and only if

R

is both zero-dimen­

satisfies both

and hence

for a

for a monoid, and

sional and Noetherian. R[S]

R

S

Thus, a.c.c.

satisfies

if and

a.c.c.

R[S]

is Artinian, then

d.c.c. and

The

on kernel ideals,

d.c.c.

on congruences.

We proceed to show that a monoid with these properties is finite.

The proof,

like that of Theorem 5.10, is not imme­

diate, but the case where

S

is a group is an elementary

result. THEOREM 20.1. d.c.c.

The group

G

satisfies both

on subgroups if and only if

G

is finite.

a.c.c.

and

272 Proof.

We need only show that

isfies both chain conditions. since it satisfies

a.c.c.

isfy

d.c.c.

Thus,

, so

direct sum of cyclic groups

G

is finite if it sat­ G

is finitely generated

G = G1©...@Gn

G^ .

Since

on subgroups, each

Z

is a finite

does not sat­

G^ , and hence

G , is

finite. THEOREM 20.2.

Assume that

T

is a cancellative semi­

group. (1)

If

T

satisfies

d.c.c.

on ideals, then

T

is a

group. (2)

If

T

gruences, then Proof. sequence

satisfies both a.c.c. T

(1):

Choose

k

of ideals of

such that

for some

(2):

Therefore T

T .

T

I

Since

Therefore I

on

satisfies

is a group.

There exists

can be written as T

is cancellative,

is an identity element of

Each ideal

pjwith respect to

T .

kt + T = (k + l)t + T = ...

u e T .

t + u

has an inverse in

on con­

and consider the decreasing

(k + l)t e kt + T = (k + 2)t + T

it follows that

that

t e T

t + T 2 2t + T 2 ...

(k + 2)t + u

d.c.c.

is a finite group.

a positive integer Thus

and

of T ,

T

T

and

t

is a group.

T induces the Rees congruence and

I <J

implies

d.c.c. on ideals, and

Consequently,

T

(1)

Pj <

Pj •

shows

is finite by Theorem

20.1 . Theorems 20.3 and 20.4 are results on congruences that could have been stated in Section 4, but they haven't been needed up to this point. will be omitted.

The proof of (20.3) is routine and

2 73 THEOREM 20.3. semigroup t

in

T

Assume that

and for

e

is a congruence on the

T , denote by

If

p

is a congruence on

then the relation t p u

p*

on T/~

is a congruence on (2)

on

If T

y

the class of

(3)

T such that

defined by

is defined by T

T/~

a y ’ b if

and

yf ^ ~

The correspondences in

(1)

an isomorphism of

the

that

the lattice of

T

satisfies

so does

lattice of

d.c.c.

(or

and if the relation ,then

y’

.

areorder-preserving.

onto

[t] p* [u] if

[a] y [b]

of each other and

p > ~

p > ~ ,

T/~ .

is acongruence on

is a congruence on

if

[t]

T/~ . (1)

y’

t

~

and

(2) are inverses p -- ►

Hence,

congruences

is

T

such

congruences on T/~ .

Thus,

a.c.c.)

p on

p*

on congruences, then

T/~ .

Part

(3)

of Theorem 5.1 and its proof establish the

next result. THEOREM 20.4.

Assume that

is a proper prime ideal

of the semigroup

T

d.c.c.

on congruences, then so does

(a.c.c.)

THEOREM 20.5. and

and let

P

U = T\P .

If the monoid

d.c.c.

on congruences, then

Proof.

Since

S

~

satisfies U .

S satisfies both a.c.c. S is finite.

S

is periodic.

be thecancellative congruence

cellative monoid

T

is finitely generated by Theorem 5.10,

it suffices to prove that and let

If

S/~ satisfies both

Thus, pick s e S on

a.c.c.

S . and

The can­ d.c.c.

congruences by Theorem 20.3, and hence is a finite group by

on

2 74 Theorem 20.2. integers

It follows that there exist distinct positive

m^

and

n^

and an element

m ls + tl = n ls + tl * some

m * n} .

Then

Let I

is an ideal of

if and only if

Then there exists a prime ideal

U

is periodic.

J n <s> = <J> .

such that

and

so the result above implies that u e U

and

such that

S

R [S ]

(2)

R

d.c.c.

Assume that J

of

S

containing

I

d.c.c.

on congruences,

there exists distinct Thus

U n I = .

fact that

We

The following conditions are equivalent.

is Artinian.

is Artinian and

R

is Artinian and

Proof.

(1) = >

(2):

Artinian ring is Artinian,

S

satisfies both

S

is finite.

a.c.c.

and

The homomorphic image of an and we have previously observed

that

S

satisfies both chainconditions

R[S]

is

Artinian.

That

(2)

implies

Condition

(3)

implies that

(3)

module over the Artinian ring

on

congruences if

is the content of Theorem 20.5. R[S] R .

is a finitely generated Thus

R[S]

Noetherian and zero-dimensional, hence Artinian. implies

(1)

is both Therefore

and the proof is complete.

The proof of Theorem 17.3

I n <s> =

on congruences.

(3)

(3)

meets

is periodic, and hence finite.

THEOREM 20.6. (1)

I

ms + u = ns + u .

u e I , and this contradicts the conclude that

S , and

for

Let U = S\J . Theorem 20.4 shows

satisfies both a.c.c.

m,n e Z+

such that

I = C t e S | m s + t = n s + t

<s>

that

s

t^ e S

20.5 is similar to that of Theorem

and indeed, (17.3) can be

used to prove the implication

275 (1) = >

(3)

in Theorem 20.6 without appeal to Theorem 20.5.

To see this, note that Artinian,

S

R [S ]

Artinian implies that

is finitely generated, and

Therefore Theorem 17.3 shows that

S

R

is

d i m R = dimR[S]

= 0 .

is periodic, and hence

finite. Theorem 20.6 is useful in determining the semigroup rings (as opposed to monoid rings) that are Artinian.

One part of

the characterization extends to the Noetherian condition, and hence is stated as part of Theorem 20.7. THEOREM 20.7. (1) and

T

R[T]

T

is Artinian if and only if

If

R[T]

is Noetherian,

is finitely generated. Proof.

R[T] ring.

be a semigroup. R

is Artinian

is finite.

(2) T

Let

(1):

If

R

is Artinian as an

then

R

is Noetherian and

The converse fails. is Artinian and

T

is finite, then

R—module, and hence Artinian as a

Conversely, assume that

is Artinian.

Let

T^

be the monoid obtained by adjoining an identity to

T .

Then

R[T°] = R + R[T] self.

Then

R[T]

.

R[T]

We consider

as a module over it­

is a submodule that is Artinian as an

R[T]-module, hence Artinian as an structure of

R[T°]

R[T^]/R[T]

= R

as

tially the same as its structure

R[T°]-module.

The

anR[T°]-module is

essen­

asa ring.

is

But

R

Artinian as a ring since it is a homomorphic image of Consequently,

R[T^]

is an Artinian

R[T^]—module —

an Artinian ring.

Theorem 20.6 then shows that

finite, and hence

T

(2):

T°

R[T]

that is, is

is finite.

By mocking the argument just given in

.

(1)

, it

276 follows that if

R[T]

rated. ring

R

is Noetherian and

isfinitely generated

is Noetherian, and hence

T

is also finitely gene­

To show that the converse fails, we show that the R[Z+] = XR[X]

group of

R

is Noetherian if and only if the additive

is finitely generated.

Noetherian, then

({rX | r e R} )

subset

°f

=i

Thus, if

XR[X]

is

is generated by a finite

{rX | r e R}

. If

r

e R , it follows

that rX = I? tk .r .X + E? ,f.r.X , i=l 1 i i=l i l * for integers r = Z^k^r^

and elements f^

, and{ r^}^

Conversely, R

k^

if

e XR[X]

is finitely generated by

is Noetherian and is generated by II .

Let

U = H

{ r^}^

+ XR[X]

is Noetherian and is generated by {r^}^ By Eakin’s Theorem, ring of of

U

since

U

is Noetherian.

R , the ideals of contained in U

XR[X]

XR[X] .

Thus

An

.

{ r^)^ > then

as a module over .

The ring

R[X]

as a module over Since

n

U .

is the prime

coincide with the ideals XR[X]

is Noetherian

R

is said to satisfy the restricted minimum

(RM— condition) , or to be an

RM— r i ng , if

satisfies the minimum condition for each nonzero ideal R .

R

is Noetherian.

The ring condition

Consequently,

generates the additive group of

(R,+)

its prime subring

.

R/A A

RM— ring with proper zero divisors is Artinian, and

a unitary integral domain

D

is an

it is Noetherian of dimension at most determines the monoid rings.

RM— ring if and only if 1 .

The next result

RM— domains within the class of nontrivial

of

277 THEOREM 20.8 is an

Assume that

S

RM— domain if and only if

isomorphic either to Proof.

If

R

Z

is nonzero. R

is a field and

or to a submonoid of

is a field, then

R [Z ]

if

R[X^]

is

Zq .

RM—domain.

is

d such that

R < R[S] e R[X^]

integral over R[S]

.

Therefore

again Noetherian and one-dimensional, hence an For the converse, R

S

S c Z Q , then Theorem 2.4 shows that there

exists a positive integer and

R[S]

is a one-dimen­

sional Noetherian domain, and therefore an Moreover,

Then

R[S]

R[S]

RM— domain.

an integral domain implies that

is a domain and

S

is torsion— free and cancellative.

augmentation ideal

I

of

R[S]/I ^ R

is

a field.

Then

is

an

R[G]

of copies of that

G- Z .

Z .

R[S]

Let G

R [Z ]

We assume that

orem 2 .6), and otherwise, S The ring localization

R Rp

be the G

is

quotient

group ofS

a finite

direct sum

is not a field, it follows

S e Z . If

itive and negative integers, then

The

is a nonzero prime ideal, so

RM—domain, so

Because

is

S

S

contains both pos- -

is isomorphic to

Z

(The­

is isomorphic to a submonoid of Zq.

is said to be locally Noetherian if each of

R

is Noetherian.

should be locally Noetherian,

In order that

it is sufficient that

Noetherian for each maximal ideal

M

of

R .

R^

R is

Each regular

ring and each almost Dedekind domain is locally Noetherian. Hence a locally Noetherian ring need not be Noetherian, even if it is a group ring that is an integral domain. a locally Noetherian ring ideal of [70].

R

R

In general

is Noetherian if and only each

has only finitely many minimal prime ideals

[10],

Theorem 20.9 lists some basic properties of locally

2 78 Noetherian rings.

In the rest of the section, we use the

following notation. and if

A

If

is anideal of

tension of

A

imbedded in

R^ .

N

is a multiplicative system in

R , we

to RN , although

THEOREM 20.9.

write A

ARN

R

for the e x ­

need not

be naturally

Each homomorphic image, quotient ring,

and finitely generated extension ring of a locally Noetherian ring

R

is locally Noetherian.

Proof. ideal of Rp N

If

A

is an ideal of

R/A , then

is Noetherian.

R

Hence

that misses

Noetherian, so

RN

T = R[a^,...,an ] R . R n M

Let .

M

and

(R/A)p/A - Rp/ARp R/A

R

is a prime

is Noetherian since

N , then

and if (r n )p r

is locally Noetherian.

P

If

is a prime

~ Rp Finally,

let

be a finitely generated extension ring of

be a proper prime ideal of

To within isomorphism,

T^

finitely generated extension ring of Noetherian ring.

P/A

is locally Noetherian.

is a multiplicative system in

ideal of

R

Therefore

T^

T

and let

P =

is a localization of a R p , hence of a

is Noetherian and

T

is

locally Noetherian. The problem of determining conditions under which a monoid ring is locally Noetherian is unresolved, but the problem has been solved for group rings. case in the rest of the section. sary conditions in order that

We

deal with this

Theorem 20.11 gives neces­ R[G]

should be locally

Noetherian, and Theorem 20.14 shows that these conditions are also sufficient.

Theorem 20.10 is a special case of Theorem

20.11 that is used in its proof.

279 THEOREM 20.10.

Assume that

Fis a field

and

that

H

is a

finitely generated subgroup ofthe nonfinitely generated

group

G

such that

G/H

augmentation ideal of

is a

F[G]

p—group.

, then

If

MF[G]^

F

is distinct

p . Proof.

H .

is the

is finitely gene­

rated if and only if the characteristic of from

M

Then

Let

be a finite set of generators for

{l-X^i}?^

generates the kernel ideal

canonical homomorphism

M F [ G = IF[G]M .

we show that 1 - X g e IF[G]m for some

for each

k e Z+ .

1 - Xp g e I . mapped to

p^ * 0 .

case where

p * char(F)

By assumption,

f = e P ^ X 18 •

Therefore

MF[G]^j

p = char(F)

{1 -X^i}™

Then

p^g e H

(1 - X 8 )f =

{ k ^ i s

that

f { M

and

generates MF[G]M . a finite subset of MF[G]M .

Without

of gene­

G/K

is a

{k^

. Since

G

is not finitely generated, then

and

G/K

contains an element

, then

f t M

. Let

onto

F [G/K] . Then

g+ K

generated J : (1 - X g )

by

generated

of order p . {l-X^i}™.

Let

J

Since

contains an element

is the kernel of

<J> .

0 = <J>(1 - X g )<j>(f)

= (1 — X g+^)(f) ,and hence

to

, which is(1 - Xg+K)p k '1

A n n (1 - Xg+K)

by G * K

be the canonical homomorphism from J

such

loss

{h^£ { k ^ , so that G

G

is the subgroup of

JF[G] m = MF[G] m

.

Assume, to

K

F[G]

is

1 - X g e IF[G]^

p— group, where

be the ideal of

f

, it suffices to show that no finite

generates

rality we assume

,

is not finitely generated in the

{1 - X g | g e G}

the contrary, that that

If

It suffices to show that

g e G .

Let

.

Moreover, under the augmentation map,

To show that

subset of

F[G] -- ► F[G/H]

I of the

by

F[G]

We have
belongs

280 Theorem 8.13. f e M

since

that

MF[G]^j

Therefore

M 2 J = ker .

This contradiction implies

is not finitely generated, as asserted.

THEOREM 20.11. p

<J>(f) e CM) , which implies that

is a nonunit of

Let

ft

R .

be set of primes

If

R[G]

p

such that

is locally Noetherian,

then the following three conditions are satisfied. (1)

R

is locally Noetherian.

(2)

G

has finite torsion— free rank.

(3)

If

then the

p— component

Proof. prove

F isa free subgroup of

^xa ^aeA

of

G

each

°f

generated by

quence of primes of R[F]

in

R[G]

Q = u~Qj

p e ft .

is satisfied.

is infinite.

A and let

To

Choose an

F be

the subgroup

Pj = Cl —

for

is an infinite strictly ascending se ­ R[F]

.

Since

R[G]

fore

G

is integral over

n . R

Q

is prime in

QR[G] q

for each R[G]

j .

and

R[G] q

does not have finite height.

is not The r e­

has finite torsion— free rank.

Let rank

such thatn RtF] = P^.

, then

Noetherian since

of

>

, there exists an infinite strictly ascending sequence

{Qj }* If

A

^xa ^aeA *

j , then { P^.}“

(1)

ro(G)

a maxim a l linearly independent

G , and assume that

infinite subset

rank

is finite for each

Theorem 20.9 shows that

(2) , let

subset of

(G/F)^

G of

r o(G) = n Assume that

such that

and let F

be a free subgroup of

p e ft and let

char(R/M) = p .

homomorphic image of

G , the ring

homomorphic image of

R[G]

Since

M

be

G

of

a maximal ideal

(G/F)^

(R/M)[(G/F)p]

is a is a

, and hence is locally Noetherian.

Applying Theorem 20.10, we conclude that

(G/F)p

is finite.

281

In order to prove the converse of Theorem 20.11, we use two preliminary results.

The first of these belongs to the

theory of abelian groups. THEOREM 20.12. the group

then

Assume that

G and that

p

is a finite subgroup of

is prime.

(1)

Gp is finite if and only ifCG/K)^

(2)

If Gp

(G/H)p (3)

is finite and if

G

Assume that

ro(G) = n of rank

finite.

finite and

Then

G

is P tains a free subgroup

and let

(1):

*

of

G

.

is finite and that

nsuch that

G

(G/F)p

con­

is

G = G © H , where H conP of rank n such that (^/F^)p =

Let

<j>

be the natural map from

be the restriction of

*(Gp) £ (G/K)p is finite if

and

(G/K)p


ker<J>* = Gp n K is finite.

is the set of

K .

is finite, then an element

If

root in

Gp

G hasexactly

form a coset

of G

P

in

pth

|Gp| G .

to

.

is finite.

G to

G/K ,

Then Hence

Gp

On the other hand,

P

H

.

Proof.

G

is finite.

is a torsion group,

is finite for each subgroup

tains a free subgroup

{0}

K

roots in

pth_

of elements of

g e G

roots,

Therefore

G

with a

for

(G/K)

p th

these roots P

is finite if

is finite. (2):

Mp = {0}

Since .

G

is a torsion group,

Moreover,

H = Hp © (M n H)

G/H = (G /H ) © (M/(M n H)) M/(M n H) (3):

is

{0}

.

, where the

Therefore

We show first that

free subgroup

F2

of

G

G = Gp © M , where , and p—component of

(G/H)p a Gp/Hp (G/F?) l p

of rank

n .

is finite.

is finite for each Thus, let

282 F* = F n F2 .

Then

F/F* = (F2 + F)/F2 c G/F2

generated torsion group,

hence afinite group.

(G/F*)/ (F/F*) * G/F , it follows from part is finite iff (G/F*)^ Since

is finite iff

Gp n F = ^

conclude that

G^

direct summand —

is a finitely Since

(1)

(G/F2)p

anc* since

that (G/F)

is finite.

(G/F)^

is finite, we

is a finite pure subgroup of say

G

G = G^ © H , where

, hence a

=

n .

By

the result of the preceding paragraph, we may assume without loss of generality that group of

H .

Since

H/F

finite, it follows from finite. that

Choose

K/(F +

F £ H .

Let

T

be the torsion sub­

is a torsion group with (2)

asubgroup

that

K

(H/(F +T))

ofH

T)=(H/(F + T))

.

n

F£ K .

since

{ Xi +

n .

we have

K

of

•

(H/F,,) = {0} v 2'p

(T + F 2^F 2^p ~ Tp =

shows that the

e

*

p—component of

Then

P

= {0} , and hence

THEOREM 20.13. with maximal ideal

.Since

H/F2

a subgroup of

G

Assume that

such that

R

H

of

, then

PR[G]p .

P n R = M

p

= {0} , (2)

By choice of

as well.

, and that

F

is

is a torsion group with no

c .

such that

is free of

is a quasi-local ring

element of order divisible by R[G]

be the

is isomorphic to that

c = char(R/M) G/F

F2

Thus, the proof of

(H/F9) = {0} L P

M , that

such that

F2

(H/F2)/((T + F 2)/F2) = H/(T + F 2) = H/K .

K , (H/K)

such

Tq(K/T)

K

K/T , and let

generated by

We show that

F + T

Moreover,

x^,x2 ,...,xn

isafree basis for

subgroup of rank

Choose

also

K/T is a finitely

generated torsion— free group,hence free. is

is

containing Then

(H/F)p

If

P

is a maximal ideal

P n R[F]

generates

283

Proof. R[F]

.

Denote by

Choose

f € R[E]

Pp

f e P .

the maximal ideal

We show that

P n R[F]

f e PpR[G]p .

for some finitely generated subgroup

taining

F , and hence

f e P £ = P n R [E] .

E

If

of

We have

of

G

con­

fePpR [E ]p

,

then

f e PpR[G]p .

Thus, we assume without loss of generality

that

G = E, so that

G/F

divisible by of

t

on

t .

c .

In this case

cyclic groups. If

is a finite group of order

is a finite direct sum

To prove the result, we use induction

G/F = (g + F)

does not divide

G/F

not

k , then

is cyclic of order R[G] = R[F][Xg ] .

k , where

c

The residue

class ring

R[G]/Pp[G] = R[F][Xg ]/Pp [Xg ] = L[u] , where v L ^ R[F]/Pp has characteristic c and u = v is a unit of L .

Thus

Since

k

L[u]

is a homomorphic image of

is a unit of

L[X]/(X

-v) . v L[X]/(X - v)

L , it follows that

is a finite direct sum of fields, and hence so is particular, {Pi>T_^

of

Pp [G]

\r

L[u]

.

In

is the intersection of the family

maximal ideals of

R[G]

.

Thus,

for

1^ j

< r

,

we have PF [G]R[G] p = (nT 1P.)R [G ] p = n ^ R f G l p = P.R[G] p . j i i i This proves the result in the case where Assume that

G/F = (g^

subgroup of

G

P n R[K]

char(R[K]/ P*) = c Since

R[G]p

F u

R[K]

that

K

Then

P* n R = M

.

be the

P* = Hence Cy = (gy + F) .

is isomorphic to a localization of

is generated by P*R[K]p*

and

•

does not divide the order of

(R[K]p*)[Cv ] , the case where PR[G]p

is cyclic.

+ F)©...©(gy + F) , and let

generated by

is maximal in

G/F

G/F

P*R[K]p* .

is generated by

is cyclic shows that By induction it follows

Pp , and hence

Pp

also

284 generates

PR[G]p

Assume that

in R

is a quasi-local ring with finitely gene­

rated maximal ideal that

R

R[G]p .

M .

A result of Nagata

[109, §31] shows

is Noetherian if and only if each finitely generated

ideal of

R

is closed in the

M— adic topology on

is the main tool we use to show that conditions Theorem 20.11 imply that THEOREM 20.14. that

p

Let

is a nonunit of

Noetherian,

that

G

This

(1) — (3)

of

R[G] is locally Noetherian. ft

be the set of

R .

primes

Assume that

R

p

is

has finite torsion— free rank

that there exists a free subgroup (G/F)^

R .

if finite for each

F

of

p € ft .

G

Then

such

locally n , and

such R[G]

that is locally

Noetherian. Proof.

Let

P n R , and let

P

be a maximal ideal of

c = char(R/M)

.

If

If

F1

of rank

c = 0 , we letGc = (0)

where

R [G c ]

.

maximal ideal

P n R[F]

M .

PR[G]p

But

R[F]

natural topology on of

R[G]

,

contained in

anc*

With ­ G= H .

is a local ring with P n R[F]

is a Noetherian ring, so

are finitely generated.

R[G]p

H

R^[G] at a

Theorem 20.13 then shows that

PR[G]p . and

R

the proof, it suffices to show that

I

R = R tG c ]

is isomorphic to a localization of

maximal ideal, we also assume that

is

(H/F^)c ={0}

is locally Noetherian by Theorem 20.9.

R[G]p

generates

Gc

R[G] = R [G c ] [H]

out loss of generality we assume that Since

M =

G = G c © H , where n such that

Then

, let

c * 0 , then

finite and Theorem 20.12 shows that has a free subgroup

R[G]

IR[G]p

To complete

is closed in the

for each finitely generated ideal P .

Thus,

take

.

285

f/h e ni= i ^1 + p^)R [G]p > the closure of where

f

R[G]

e

and

rated subgroup

E

h e R[G]\P .

of

G

IR[G]p

Let

IE = I

I = IgR[G]

n

containing

R [E]

by choice of

F

Theorem 20.13.

and let E

such that

R[G]P

P£ =

and

belongs to

n R [E] .

PR[G]p = PgRtGjp

p e )r [g

]p i n

is faithfully flat as an

Then

by

R [E]D FE

is contracted from

r

[e ] p e •

R[E]D —module, each *E

F

ideal of

P

I

f,h,

Hence

f/g * rn± = i CI e + Since

and

R[G]p ,

Choose a finitely gene­

each element of a finite set of generators of R [E] .

in

R[G]D .

Therefore

v

f/g « n“ (IE + p|)R[E]p

, E

the closure of R[E] p *E

Ip.R[E]D h *E

is a local ring,

in the natural topology. IpR[E ]D h

£ IR[G]p .

f/g e IER[E]p

Since

is closed, and hence E

This shows that

IR [G ] p

is closed

E in the natural topology on

R[G]p , thereby completing the

proof of Theorem 20.14. As an application of Theorem 20.14, we determine those group rings that are almost Dedekind domains.

Recall that an

almost Dedekind domain may be defined as a locally Noetherian Priifer domain (see the remarks at the end of Section 13) . Thus, the following result is a consequence of Theorems 13.6 and 20.14. COROLLARY 20.15. Dedekind domain if satisfied.

The group ring

R[G]

is an almost

and only if the following conditions are

286 (1)

R

is a field,

(2)

G

is a subgroup of

(3)

either

each

char(R) = 0

(some) nonzero element

integer

k(g)

such that

Q , and or

ch arR = p * 0

and for

ge G , there exists a positive

g/p^^

{ G .

We note in particular that Corollary 20.15 is strong enough to show that

R[G]

almost Dedekind domain.

need not be Noetherian if it is an For example,

Q[Q]

is such a domain.

Section 20 Remarks Additional material on Artinian rings can be found in [136, Section 2 of Chapter IV]. [51] concerning

RM— rings.

See

[33] and Section 40 of

The papers

[10] and [70] treat

locally Noetherian rings in general, but the material on locally Noetherian group rings is more closely related to the papers

[90],

[23], and [52].

In particular, Lantz in [90]

also determines conditions under which

R[G]

is locally

Cohen-Macaulay, locally Gorenstein, or locally regular

(not

in the sense of von Neumann); he also considers briefly the problem of determining when

R[G]

satisfies the chain con­

dition or the second chain condition on prime ideals

(see

also [97]). The proof of Theorem 20.12 uses the fact that direct summand of

if

G

is a

is finite. This is a conseP quence of a more general result due to Kulikov — see Theorem 27.5 of [47].

G

Gp

CHAPTER V DIMENSION THEORY AND THE ISOMORPHISM PROBLEMS

Chapter V contains material on two topics with little overlap.

The first of these topics is that of the (Krull)

dimension of

R[S]

21.4 states that S

, which is treated in Section 21. dimR[S] = dimR[G]

with quotient group

G , and

for a cancellative monoid

dimR[G]

is determined in

terms of an appropriate polynomial ring over 17.1.

Theorem

R

by Theorem

The other sections of Chapter V concern the isomor­

phism problems for semigroup rings.

More specifically,

Sections 23 and 24 deal with the question of whether isomor­ phism of

R^tS]

and

R 2 [S]

implies isomorphism of

R 2 , and Section 22 considers

R— automorphisms of

basic tool in the development of Section 24.

R^ R[S]

and , a

A survey of

some results concerning the other isomorphism problem --whether isomorphism of R[S^]

and

R [S 2 ]

S^

and

S2

follows from that of

--- is given in Section 25, but no proofs

are included.

287

288 §21.

Dimension Theory of Monoid Rings

In this section we seek to determine the (Krull) dimension of a monoid ring general monoids

R [S ] . S

While some results are known for

(for example, see Section 17), the theory

is complete only in the case where

S

that is the case we treat.

cancellative, we show that

dimR[S]

= dimR[G]

For

, where

G

S

is cancellative, and

is the quotient group of

S .

By Theorem 17.1, this reduces the problem to the study of the dimension of a polynomial ring

over

R > a

topic which has been the subject of intensive investigation. It turns out that even the proof of the equality dimR[G] of

dimR[S] =

depends heavily on the known prime ideal structure

R[X^,...,Xn ] .

Thus, to begin the section, we review

some of the notation and results from this theory. for the statement that we label as part

(4) of (21.3), all

the results cited can be found in [51, §§30, [137, Chapter VII, §7]. terminology.

R[X^,...,Xm ]

and for an ideal A[X^,...,Xm ] . of

R^m ^

A

of

in

R , we write m

R , A ^

A chain

the following statement Special Chain Theorem;

(21.1)

R ,

of prime ideals

(P^ n R) ^ 0

and

k .

is a member Part

(2) of

(21.1) is usually known as Jaffard's the labeling of the parts is signif­

icant in the sense that follows from

between

for

= A*R[X^,. . . >Xm ] =

Pq g P^ c...c P^

i

R^m ^

indeterminates over

is said to be special if

of the chain for each

31] or in

First we introduce some notation and

For a unitary ring

the polynomial

Except

(2)

follows from

(1)

and

(3)

(2) .

Assume that

R^m ^ is finite— dimensional and that

289 Q

is a proper prime ideal of

let

(M.K (1)

ht(Q)

.

A Ae A

.

Let

P

= Q

be the set of maximal ideals of

nR

and

R .

ht(Q) = ht(p(m ))+ ht(Q/P^m ^) ; in particular,

can be realized as the length of a special chain of

primes of (2)

R^m ^

with terminal element

dimR^m ^

can be realized as the length of a special

chain

Pq < P^ <...
over,

PR n R (3)

Q .

of proper primes of

is maximal in

R

R^m ^ .

Mor e­

for each such special chain.

d i m R ^ = sup ( h t ( M ^ )

+

•

We subsequently show that (21.1) extends to each monoid ring

R[S]

such that

S

is torsion— free, cancellative, and

finitely generated. A finite— dimensional unitary ring

R

is said to satisfy

the saturated chain condition if any two saturated chains P < P^ <...< Pr = Q

and

P <

<...
fixed primes have the same length.

between two

Affine domains over a

field satisfy this condition. (21.2) the field

Let k .

Let

D = k[x^,...,xn ] d = tr.d.(D/k)

zero proper prime ideal of

be an affine domain over

and let

P

be a non­

D .

(1)

ht(P) = d - tr.d.[(D/P)/k]

(2)

dim D = d .

(3)

D

.

satisfies the saturated chain condition.

Unlike the parts of (21.1) and (21.2), the first three parts of (21.3) are not closely related to the fourth; we state the four parts together to avoid repetition of hypo­ theses .

290 (21.3)

Assume that

tient field (1)

is a unitary domain with quo­

and let

be a finite subset of

<J>:D^--- ► V [ t 1 , . . . ,tm ]

If

defined by height

K

D

K .

is the homomorphism

(f (Xx ,. . . ,Xm )) = fCt1 ,...,tjn) , then

ker (*) has

m .

(2)

If

D

admits an overring of dimension

then there exist

u^,...,ur _^ e K

dim D[u^, . . . ,ur _ ^ (3)

If

D

eachoverring of (4)

such that

r .

is an D

r > 1 ,

n— dimensional Noetherian

has dimension at

most

domain,

then

n .

d i m D ^ = m + sup{D [u.^, . . . ,um ] |

•

A proof of part (4) of (21.3) can be found in [41]. the rest of thesection, we assume that S

is a cancellative monoid, and

S .

G

R

is a

In

unitary

ring,

is the quotient group of

Although its proof will be delayed, we state the main

result of the seciton at once. THEOREM 21.4. To prove

dim R[G] = dim R[S]

.

(21.4), we establish a series of results that

have the effect of showing that it is sufficient to prove the result in cases where various restrictions are imposed on and

S .

The first such reduction is immediate:

is a quotient ring of and hence

R[S]

(21.4) holds if

that is, if

dim R = °°

, then R[G]

or if

dim R[G] s dim R[S]

G

R[G] ,

is infinite-dimensional — r o(G) ” 00 •

(21.4), we need only consider the case where finite— dimensional and

since

R

Thus, R

in proving is

has finite torsion— free rank.

Theorem 21.5 provides another reduction.

291 THEOREM 21.5.

If the equality in (21.4) holds for a

finite-dimensional unitary integral domain and a finitely generated torsion— free monoid, then it holds in general. Proof. and

To prove the equality of (21.4) for general

S , we may assume that

that

G

is finite— dimensional and

has finite torsion— free rank

subgroup of

G

of rank

k .

be the quotient group of integral over

R [ S f]

R[F']

if

.

R

Thus,

n e Z+ .

a ,b e S .

Let

S1 .

and that

if

We choose

ng = na - nb e F'

R[G] ns e

suffices to prove

G

We have

RTS]

F' is

S 1= F n S

for some

g = a - b

for some

na , nb e S ! .

Then

dim R[S] = dim R [ S !] S

with

S'

and

, it

S

is

is finitely generated. dim R[G] < dim R[S]

Let

P = P q a R , then

and let

the equality in the case where

Suppose that

prime ideals of

be a free

is integral over

such that

dim R[G] = dim R [F 1] , so by replacing

torsion— free.

F

, and this establishes the two statements

concerning integrality.

torsion— free and

Let

S’ = S n F

g e G , then n e Z+

k .

We observe that

s e S , then

Moreover,

R

Pq < P^ <...< P^

R[S] P[S]

of length

P[G]

S

is

be a chain of proper

t > dim R[G]

is prime in

and the same result shows that

, where

R[S]

.

If

by Corollary 8.2,

is prime in

R[G]

.

Moreover, dim (R/P)[S] = dim R[S]/P[S] and

D = R/P = P^/P[S]

1 £ i £ t . G

and write

* t > dim R[G] ^ dim (R/P)[G]

is a finite— dimensional unitary domain.

Let

for

^or

Let

each

i, and choose

{g.}1? ^ be a finite = s i “ ti * with

€ Qj. ~ ^i-i set of generators

s^ , ti e S

for each

for i .

292 Let

T

(s i }^

be the submonoid of ft.}™ u Supp(f1) u.

u

generated,

G

i .

generated by

.uSupp(ft)

is the quotient

dim D[T] ^ t > dim D[G] for each

.

S

torsion— free of rank

, for

S

over

generates

Since

S

If

expressible in the form

GN

M e Z+

, where

has quotient group R[T]

R[xJ*,...,X**]

pure monomials r

Q—vector space,

If

{tj}^=1 t^

is a is uniquely

e Q

.

It

such that for each

T

u { M t j , and hence

finitely generated over

as a

S , then each

Mtj e Zs1+...+Zsn = Zs^©...©Zs^

we're dealing with in

is the set of non­

^i=iajisi > where

follows that there exists

R[T]

N

.

finite set of generators for

subring of

is

is ann— dimensional vector space

contains a basis

is integral over

G

n , we can reinterpret the equality

G^

T

D[T])\CQ^_1 nD[T])

is finitely generated and

zero integers, then

over,

T , and

f^ e (Q^ n

asfollows.

by

is finitely

group of

dim R[G] = dim R[S]

S

T

This establishes Theorem 21.5.

In the case where

Q .

Then

.

j .

Then

R[S]

is the monoid generated

dim R[S] = dim R[T] Zs^©...©Zsn = G .

; more­

Thus, what

, to within isomorphism, is a containing

R[X^,...,Xn ]

R[X^,...,Xn ]

that is

by a set

h. = X?l...Xan , where each 1 1 n

aj

is an

integer (not necessarily nonnegative); we note that this use of the term ’’pure monomial" is a extension of that used in Section 15, where the exponents negative.

In this notation,

assertion that Since

a^ were assumed to be n on ­

(21.4) is equivalent to the

dim RfX*1 ,. . -X*1 ] = dim R f X ^ . . . ,X ,1^,. . .,hk] .

dim R[Z^n ^] = dim R^n ^

by Theorem 17.1, we have

established the following result.

293 THEOREM 21.6.

The equality in Theorem 21.4 is equivalent

to the following statement.

If k

unitary domain and if

D

is a finite— dimensional

is a finite set of pure

monomials in the indeterminates

over

dim D[Xr ...,Xn ] = dim DfXj, . . . ,Xn ,(h. }J]

D , then

.

We turn to the proof of the statement in (21.6).

The

first step in the proof is a result about polynomial rings. THEOREM 21.7. P

Let

D

be a unitary integral domain,

be a proper prime ideal of

integers such that ht p^-k) > m + i

Proof. ht

1 < m < n.

for

By passage to ideal

ht p^m ^ £ m + 1 .

m > ht P = dim D . dim

Part

= n + ht p^n ^ >

case that where

dim

K

If

ht

m

P . Thus,

Since

+ 1 , so part

= m

dim

be

> m+ 1 , then

If

D

is

ht P >m + 1 , then

assume that

(3) of (21.1) implies that n + m + 1 .

is the quotient field of

dim

n

But it is also the

= n + sup{dim D[t^,...,tn ] |

dim D[s^,...,sm ] > m Then

and

D p , we assume that

D , by part

Therefore, there exists an overrring of least

m

itsuffices toprove that

quasi— local with maximal certainly

, and let

m< k £n .

Clearly

> m + 1 .

D

let

(2) of (21.3)

+ 1

D

> (4) of (21.3).

of dimension at

implies that

for some subset

+ sup {dim D[t^, . . . ,tm ] |

K • = K } > 2m + 1 .

= m + ht p^m ^ , it follows that

ht P fm^ > m + 1 . If

D

polymomials

is a unitary integral domain, the set f e

B

of

of unit content forms a multiplicative

294 system in

, and the quotient ring

is denoted by

D(X^,...,Xm ) .

of

D(X1 ,...,Xm )

If

Q

.

is a prime ideal of

QD(X^,...,X ) if

Assume that

containing

exists a unique prime

P

n

J

.

Q

P^m ^

of

J

Domains of the form

J

T

such that

result applies to the domains

Let

D

Assume that

D

, there

Q ,

namely Q =

In particular, a

= p(m )

n

, where each

h^

#

is a

P

of

of part

and

Moreover,

D^m ^

J

is

of Theorem 21.6. (1)

of (21.1).

B be as in the preceding

in

is a finitely

D(X^,...,Xm )

and that each

extends to a prime ideal of J .

proper prime ideal of

PT

D , so the following

J = D^m ^[t^,...,tn ]

generated extension of prime of

P^m ^

D^m ^[(h^}]

Theorem 21.8 is a generalization

paragraph.

B

> satisfy the condition that

for each prime ideal

THEOREM 21.8.

is a subring

D , then there exists

s DfX*1 ,....X*1] s D(X1 .... Xm ) .

prime in

of

= (D^m ^)g .

ht Q = ht Q .

T = D^m ^[(hj}^]

pure monomial in D (m) = T

Thus

lying over

Therefore

of

J

that misses

is a proper prime ideal of

unique prime

(D^m ^)g

and if P = Q

n

D ,

If

Q

then

is a

ht Q =

h t ( Q / p W ) + ht( P(m^) 2 m + h t ( P (m)) .

Proof. then

Observe that since

PJ = P ^ J

P (m) = p(m)JB n J = P ^ J

We establish first the inequality

is prime in J

for each prime htCQ/P^J)

P

s ® •

of

D.

Thus,

to within isomorphism,

(D/p)(m) = D(m)/pOO s j/pMj £ ( D ^ ) b / P ^ ( D ^ ) b , and under this identification, N

(Q/P^m ^J) n (D/P) = (0)

is the set of nonzero elements of

D/P

.If

,it follows that

,

295 k ^

c (J/P^m ^J)N c (k^m ^)g* , where

field of

D/P

and

B*

(3)

is the quotient

is a multiplicative system in

Since each overring of part

k

k^m ^

has dimension at most

of (21.3), we conclude that

k^m ^

.

m . by

ht(Q/P^J)N =

ht ( Q / P ^ J) £ m . The inequality

ht Q £ ht(P^m ^J) + ht(Q/P^m ^J)

clearly holds; we establish the reverse inequality by in­ duction on

ht

Q . If

or

.

In either case, the result holds.

P = (0)

ht Q = k > 1

ht Q = 1 , then either

and let

of prime ideals of

(0) <

J .

Set

<...<

< Q

P.^ =

n D .

then the induction hypothesis implies that ht Qk _1 + 1 = ht P (m)j + ht P ^ J

+ ht(Q/p(m )j)

r = ht(Q/P^m ^J) have

.

be a chain If

.

P = P^ ,

ht Q = k =

Thus, assume that

k - 1 = s + ht P^m ^J . J

Assume that

ht(Qk-1/P^m ^J) + 1

, s = ht(Qk l /P^m ^J)

k = ht Q > r + ht

Q = P^m ^J

P1<

< P

By assumption we

Therefore

s + ht P^m ^J + 1 =

> r + h t ( P J/?1

J) + ht P^m)J .

Clearly then, we need only show that

r + ht(P^m ^J/P^m ^J) £

s + 1 ; this inequality is patent if

r > s , and hence we

assume that Let

s > r .

<J>:D ^

[Y, , . . . ,Y ] -- ► J 1 m

momorphism determined by denote by

U

the kernel

is to show that prime ideal diagram.

andset

H

be the canonical

<|>(Y^) = t^ of

<J> .

for each i

D .

,

The next step in the proof

ht(H^m +n ^ + U)/H^m+n^ = n of

D ^ — ho-

for each proper

Consider the following commutative

and

296 (D/HJIXj,.

J - D[X,, .. ..JL.t.

. •,Xm ,Y1.... Yn ]

(D/HJtXj, . . . . X ^ t *^),...* (tn)]

( D ( m ) ).

Here (D/H)

is the canonical map.

t

(m)

and

0 ^

t

duces an epimorphism

[(D/H) (m:)]

t

of

to

since

t*

of

x(D^m ^) = Hence

n B =

(m) (D

(D/H) (X1 , .

(B) J

(B)

We note that

(D/H) [Xr . . . ,Xm ,T* (tx) , . . .

J

,T* (tn ) ]

.Finally,

(D/H)(m+n)

relations

; we note that part

ker a

has height

H (m)(D (m))

n J = H (m^J .

kernel ^ ( H ^ J ) it follows that fore

T(H^m+n^ + U) has height

(D/H) (m+n)

obtain the desired equality

Consider a chain

of prime ideals of .

primes of first to

Then

of (21.3)

The kernel of

is

has

Considering

as

is the kernel of n.

and

Finally,

a .

an

,

Th ere­

x(H^m+n^ + U)

D^m+n^

jjCm+n)^(m+n) ^ fTOm which we

ht(H^m+n^ + U)/H^m + n ^ = n .

We return to the proof that

4>_ 1 (Q)

(1)

(H^m+n^ + U)/H^m+n^ under the canonical iso­

morphism between

s + 1 .

is the

determined by the

Therefore

= H^m+n^ + U .

x(H^m+n^ + U)

corresponds to

n .

ij;

onto

( D / H ) h o m o m o r p h i s m of

shows that

onto

,Xm ) , and the restriction

is an epimorphism of

a(Y^) = T *(t^)

,X ) ’ nr

) B = D(Xj,.

in-

t

r

+ht (P^m ^J/P^

p (m+n) < U, 1 of length

J)

^

<...< U =P (m+n) n

+ U

n , and let

p(m+n) < Uj <...< Un c M

D^m+ n^ , each meeting

D

in

(D(m+n); p (m+n)} = (D/p)(m+n)

M =

is a chain of P .

By passage

and then tQ

k (m+n)

^

297

where

k

is the quotient field of

D/P , it follows from

(21 .2) that this chain can be refined to a chain

p (ra+n) < „

<

< y

of length Since

=

p (m+n)+ u < n

n

1

<...< n+l

u . = M n+w

n + w = ht(M/P(m+n)) = n + ht(M/Pfm+n) + U)

p(m+n) + u

the preimage of

P^m ^J

under

.

<(> ,

it follows that

ht (M/P^m+n^) = n + ht ( Q / P ^ J)

=

n + r s n + m .

Similarly,

> then

if

M^j

=
it

M k-1 n D = P1 , ht(Mk _1/(pJm+n;) + U)) = htCQk .1/p{m)J) = s , and

ht(Mk ^/P^m+ n^) = n + s < n

Applying

(1)

+ m;in

particular,

s s m .

of (21.1) to the polynomial ring

(D/P^) [X^, .. . ,Xm ,Y^, . . . ,Y ] ., we conclude

that

ht(M/p|m+n)) = ht(P Cm+n:)/p{m+n)) + ht(M/P(m+n)) ; hence n + r + ht(p(m+n)/pfm+n)) = ht(M/P^m+n)) > 1 + ht(Mfc_i/P^m +n^)

= 1 + s + n.

Thus, we

have

>l + s -

r .Since

s > r ,

ht(P(m+n)/p(m+ n)) 21.7 shows that

ht^^/P^^)

s - r < t < n + m .

Now

^ 1 + s - r

s- r + 1 .

ht(p(m )j/p^j)

s - r + 1 also,

the desired inequality

for

s - r < s < m , so we have

h t ( p / p ^ m ^) > >

Theorem

But then and this is equivalent to

r + ht (P ^

J/P^,

J) > s +1.

completes the proof of Theorem 21.8. Proof of Theorem 21.4. ment in Theorem D^n ^[{hj}^]

If

We prove the equivalent state­

21.6 concerning Q

D^n ^ and

J=

is a proper prime ideal of

J and

if

This

298 P = Q n D , then Theorem 21.8 implies that h t ( P (-n -lJ) + ht(Q/P(n)J) s h t f P ^ J )

ht Q =

+ n < dim

.

This inequality is sufficient to imply the inequality dim J ^ dim

, and the reverse inequality has already

been noted. fk)

J

The prime ideal structure of

has clearly been

used in a strong way in the proof of Theorem 21.8, and hence in the proof of (21.4).

On the other hand, the part of the fkl Rl J

theory pertaining strictly to

in the proof of (21.4)

is confined to (21.1) and to Theorem 21.7. extends part

(1)

Theorem 21.8

of (21.1) to monoid rings

is a finitely generated monoid lying between

D[T]

, where

T

ZQS^@...©Zsn and

ZSi@...@Zsn . We proceed to extend each part of (21.1) to mon­ oid

rings

sion— free. chain in R[S]

R[S] , where

S

is finitely generated and tor -

To accomplish the extension, we define a special

R[S]

as a chain

Pq g P^c...cP^

with the property that

chain for each

i

between

THEOREM 21.9. that

S

of prime ideals of

(P^ n R) [S] and

0

Assume that

is a member of the

k .

R

is finite— dimensional and

is torsion— free and finitely generated.

a proper prime ideal of

R[S]

ht Q = ht(P[S]) + ht(Q/P[S])

and let .

Let

P = Q n R .

Moreover,

ht Q

Q Then

can be

realized as the length of a special chain of primes of with terminal element Proof.

be

R[S]

Q .

As in the paragraph following the proof of

Theorem 21.5, there exists a finitely generated submonoid of

S

such that

S

is integral over

basis for its quotient group

, and

T , T

contains a

G^ = G - Zn

for

T

299 some

n e Z+ .

Let

= Q n R[T] .

ht(P[S]) = ht (P[T]) , that and that ht(Q) = ht(Q^) multiplicative system

We show that

ht(Q/P[S])- ht CQX/P[T]J

.

Since

P[S]

does not meet the

{Xs | s e S} , we have

ht(P[S]) = ht(P[G])

= h t ( P ^ ) ;by the same argument,

h t (P[T]) = h t ( P fn))

, and hence ht(P[S])

Qf = Qj/P[T]

.

9 it: suffices

ht Q* = ht

In that case we have

R[S]

and

ht Q* = n -

k 2 = tr.d.

[R[T]/Q*]/R .

ht Q* .

k^

overR , and Since

follows that

R

is a field.

ht

R[S]/Q*

ht Q* =

Q-. = ht(P[T]) +ht Q* .

1

W 0 < Wj^ <...< Wv l < length prime in

v = htfQ^) R[T]

domain case,

.

1

be a chain of primes Let

Pq = W q n R .

, we necessarily have it follows that

Consequently,

in R[T]

Since

W Q = P q [T]

= ht(P[T])

But since

ht Q < ht

assertion in (21.9) that

Pq [T] .

and

ht Q .

If

the statement holds.

ht

of is

From the

R[S]

is integral over

also holds.

ht Q = ht(P[S]) +

Q = 0 , then

In the case where

<

+ ht Qj =

To prove the statement concerning special induction on

is

v = htCQ^/PQfT]) =

ht

ht(P[S]) + ht Q* < ht Q . , the inequality

R

general case, let

ht (P [T] /P q [T] ) + ht (Q1/ P[T ]) £ ht(P[T]) + ht CQ ^PfT]) ht Q 1 = v .

and

is integral over

establishes this result in the case where

an integral domain. In the

R

is the transcendence

k^= k 2 , and hence

We next prove that

Theorem 21.8

the

and

ht Q*= n — k 2 , where

1

R[T]

Let

R[T] are affine domains over

, where

R[S]/Q*

, it

.

t0 establish the

in the case where

degree of

R[T]/Qji

= ht(P[T])

By passage to(R[S]/P[S] ) [(R/P)\{0 )]

(R[T]/P[T]) j (R/p)\{o}] equality

,

Q =

This proves ht(Q/P[S]) . chains, we use (Q n R)[S]

ht Q = v > 0 ,

300 the equality

v

=

exists a chain

ht(P[S])

<- ht(Q/P[S])

Q q < Q^ <...< Qv .^ < Qy = Q

length

v

Q- n R

for each

such

that Qu = P[S]

for some

Qu _i

has height

of primes of

u < v .

i ,then the given chain

the contrary case it follows that ideal

implies that there

If

P =

is special.

(Qu _i n R) < P .

In The

u - 1 < v , and hence the induction

hypothesis implies that there exists a special chain Qg <...< Q*.^ = Qu _i •

Since the chain

Qg <...< Q*_i < Qu <...< Qv = Q

is also special,

this com­

pletes the proof. THEOREM 21.10. and that

S

Assume that

is finite— dimensional

is finitely generated and torsion— fre e.

n = dim R[S]

Po < P1 < ‘* *< Pn

over,

Pn n R Proof.

P roP er primes of

is maximal in We have n = ht M

R

R[S]

.

More­

for each such special chain.

for some maximal ideal

M

of

, and hence the statement concerning the existence of

special chain of length

n

follows from Theorem 21.9.

prove the statement concerning maximality of P^ =

Then

can be realized as the length of a special

chain

R[S]

R

(Pn n R)[S]

special chain

.

By passage to

Pn n R , let

(R / pn n R )[S]

, we can assume that

with quotient field is to prove that

F

and that

R = F .

of proper primes of

F[S]

m , where

.

m = TqCG)

dim R^m ^ > m

To

an(* the

(Pn n R)[S]/(Pn n R)[S] < ...
(R/(pn n R))[S]

in

is an integral domain

Pn n R = (0)

•

Our object

Since the chain extends to a chain , it follows that

On the other hand,

+ dim R , so

THEOREM 21.11.

R

a

dim R = 0

Assume that

R

and

n < dim F[S] = dim R[S] = R = F .

is finite— dimensional,

301 that

S

is finitely generated and torsion— free, and that

rg(G) = m .

Let

be the set of maximal ideals of

R . (1)

If Q

is a

Q n R , then (2)

R^

.

dim R [ S ] = supCdim R^ [S] A (1):

Set

P =

P^ =

nR^ .

R^[S]

Since

shows thatht(Q^/P^[S]) = m Theorem 21.9

A = m + sup{ht(MjS]) }

R 1 = R p , Pj = PRp > and

Then is maximal in

, and

and if

ht Q = ht(P[S]) + m .

Proof. QRp[S]

maximal ideal of R[S]

shows that

, P1

R ]/Pi

=

is maximal in is a

field,

= dim CR-^/P^) [S ] . ht Q = ht

A .

(21.2)

Then

= ht(P^[S]) + m

=

ht CP[S])+ m . (2): ideal R[S]

M .

Assume first that

Choose a special chain

of length

n = dim R[S]

Pn

n R = M ,andthen

ht

P n

R is quasi— local with

=ht(M[S]) +

part m.

m + sup^thtCM^[S])} ,

(1)

Pq <

< • • #< PR

. Theorem 21.10 shows implies that

It follows that and it

maximal

that

dim R[S] = n =

sup,{dim RM a

[S]}

=

is clear that

dim R[S] > dim RM [S] for each A . On the other hand, mA Theorem 21.10 implies that dim R[S] < dim RM [S] for some mA A , and this completes the proof.

Section 21 Remarks The treatment in this section follows closely that of Arnold and Gilmer in [12].

At the same time, we have omitted

proofs of some cases of (21.4) that are derived in [12] w it h­ out using (21.7) or (21.8); these include the cases where

R

is a Noetherian ring, a ring of nonzero characteristic, or a Prufer domain.

The proof in the case of a

302 characteristic uses the Noether Normalization Lemma in an \r

affine domain k .

k[ X^,...,Xn ,{h^}^]

over a finite prime field

Arnold and Gilmer recognized that an analogous proof in

the case of characteristic

0

could be obtained if the

normalization lemma could be extended to

Z [X^,...,Xn ,{hj}^]

.

Using a theorem of Shimura, Moh [105] subsequently proved the required normalization lemma over

Z .

An extra dividend of

the proof using (2 1 .8) is that it essentially yields as corollaries the extra information about the prime ideal structure of

R[S]

contained in Theorems 21.9,

21.10, and

21.1 1 . While the equality

dim R[S] = dim R[G]

is valid for

an arbitrary cancellative monoid, an example in [12] shows that the hypothesis that necessary in (21.9),

S

is finitely generated is

(21.10), and (21.11), even if

free of finite rank and

R

is a field.

G

is

303 §22. Let

R

be a unitary ring and let

automorphism (r) = r R[S]

<J> of

for each

R[S]

r e R .

.

The set of

AutRR[X,Y]

be a monoid.

An

R— automorphism

if

R— automorphisms of

it is denoted by

Not much is known about

For example,

R[S] S

is called an

is a group under composition;

AutRR[S] S .

R— automorphisms of

AutRR[S]

AutRR[ZQ ] - AutRR[X]

for general

is known, but

is not known, even for a field

R .

Group rings

are more tractable in this regard, due to the fact that a ring automorphism induces an automorphism of the unit group of the ring.

Thus, in Theorem 22.4 we determine

We begin considering the polynomial ring If

y

is an

zi=o8ixl e Thus

y

R-endomorphism of

* we have

is an

X — ► y(X)

•

.

map

R— endomorphism, which we denote by

f

AutRR[X]

in order that

is y^

is

Theorem 22.1 resolves this question.

THEOREM 22.1.

Assume that

f = E^=Qf iX i e R[X]

map g — ►g(f)

g(X) =

= g(p(x))

Therefore the problem of determining

a bijection.

, then for

, then the substitution

that of determining conditions on

that

R[X]

^(sOO) =

f(X) e R[X]

g(X) — ► g(f(X)) y^ .

R[X] .

is the substitution determined by

Conversely, if

AutRR[Zn ] .

.

.

R

Let

is a unitary ring and yf

be the substitution

The following conditions are equivalent.

(1) y^

is an automorphism.

(2)

y^

is surjective

(that is,

(3)

f1

is a unit of

R

and

fi

R[X] = R[f])

.

is nilpotent for

i > 2 . Proof.

The implication

(1) — > (2)

is clear.

Assume

304 that if

(2) a

is satisfied.

If

P

is a proper

is the canonical map of

(R/P) [X] = (R/P)[a(f)]

.

R [X ]onto

Since

that

a(f)

a^ e P a^

deg a(f)

.

has degree

for each

, then

R/P is an integral

domain,

imply that

(1): y^

assume that show that

1 .

X

(R/P)[a(f)]

e (R/P)[a(f)]

Consequently,

i > 2 , so that

is nilpotent for (3) = >

Since

and

(R/P)[X]

the degree of any nonzero element of multiple of

prime ofR

a^

is a

, it follows

a^ {

P

while

is a unit of

R

and

i > 2 .

We show that the conditions given on

is both injective and bijective.

g(f) = 0 .

To show that

f

Thus,

g = 0 , it suffices to

g(X - f Q) = 0 , for the substitution mapping

yY r is clearly an automorphism of R [X ] . Hence, we 0 assume without loss of generality that fg = 0 . Let g(X) = Z ^ g . X 1 . Assume that

The constant term of

g(f) is

= ... = g^ = ^ = 0 , where i £ m

g(f) = f1 Cgi + g i+1f +•••+ gm fm ’1) • however, that g- + g-_Llf -i 6 i+l

f

follows that

g = 0 , and

and g. = 5i y^

R[X]

assume without loss of generality that

k , the order

of

R .

If

We assume that the inclusion element

f

where

r >

where

B^

e R [X ] 2. has

fg = 0

R[X] c R[f]

f = X A[X]

it

R[f] =

y^ , we can

of nilpotence of the ideal

k = 1 , then

0 =

, and hence

Since

R[fj*(f - fq)] , in proving surjectivity of

on

Then

0 .By induction 7

is injective.

We then establish the inclusion

0 .

Corollary 8.6 shows,

is not a zero divisor in

+. . .+ g fm_1 = 0 5m

.

gQ =

and

f^ = 1 .

by induction B^ = (f2 ,...,fn )

and the inclusion is clear. c A [f ]

of the desired form

for

holds for each which k < r ,

Consider h = X + h 2X 2 + '**+ hvXv € R[X] order of nilpotence

r . For i

>2 , note

,

305

that

h^h1

has the form

v ij 6 ^Bh ^2 * has the form

T h u s > if X

=

, where each

g = h “ h 2h2 “ • • '“ hvhV ’ then

+ g 2^ 2 + -• • + gm Xm » where each

It follows that

Br l c B?^r ~ ^ g “ h

hypothesis implies that R [g ] s R[h]

h^X1 +

R[X]

e

g

(B^)2 .

= (0) , so theinduction

c R[g] .

Since the inclusion

is clear, this completes the proof.

Some of the ideas in the proof of Theorem 22.1 are u s e ­ ful in determining case

n = 1

AutRR[Zn ]

as well.

We consider the

separately, not because it is necessary to do

so, but primarily because the notation is much simpler in this case.

Thus, we regard

R— automorphism



of

R [Z ]

R[Z]

as

maps

X

R[X,X *] to a unit

R[X,X ■*■] , and is completely determined by (X) = u , then u e R[X,X"*] map

^(Ea^X1) = Ea^u1 .

is a unit, then the

f(X) — ► f(u)

of

R[X]

R[u]

has a canonical

R[X] ^ i y» = R[X,X

Ea^u*

f(X) e R[X,X *] .

R[X,X"*]

that maps

of (22.1).

X

of

Conversely, if

, and the extension maps

$ the

R[X,X *] , we denote by

u

u ; to wit, if

R[u ]|ui^«> = R[u,u *] for arbitrary

An

substitution

onto

extension to a homomorphism of

.

onto

f(X) =

For a unit

Ea^X1 to

u

R— endomorphism of

to u .

Theorem 22.3 is the analogue

The proof of Theorems 22.3 and 22.4 use the

following auxiliary result. THEOREM 22.2. prime subring of the ring ntu.u'1 ,!!] .

Assume

n . R .

Let

Then

that u be a

u — n

R

is a unitary ring with

unit and

n

be a nilpotent

is a unit of the ring

of

306 Proof.

Assume that

= 0 .

s.. „ w rk-l i k-l-i. (u - n ) C Z i= Qu n ) , so

which is in

IT[u"^,n]

THEOREM 22.3. that

u =

R— endomorphism of

u^ = u ^ — n^ =

f

.-1 „k-l i - k k - i -1 (u - n) = £i = 0u n

,

.

Assume that

=

Then

R

is a unitary ring

is a unit of R[X,X

. Let

R[X,X~*]

X

that maps

<J>u

to

and

be the

u .

The

following conditions are equivalent. (1)

u

is an automorphism.

(2)

u issurjective (that

(3)

ui is

Proof. follows from We write

nilpotent for

(1) <=*=>(2): (2)

.

generated by

r^ e R^

and hence (2)

where

u^ .

R ^ X j X -1] = R ^ U j U " 1] .

Noetherian ring

let

a

2i=mriul = ®

X =Za^u1 , X~* = Zb^u* , u -1 = Zc^X^ R

$u =>

R^[X,X"*] for each

Then

.

.

We need only prove that

cients r^ , a^ , b^ , c^ , and

Since

R[X,X *]= R[u,u *])

i { {- 1,1}

Assume that

the unitary subring of

and

is,

(1)

9 where

r. e R .

. Let

R^ be

1

and the coeffi­

R^

is Noetherian

The restriction of

<J>u

to the

issurjective, hence injective. i , it follows that each

r^ = 0 ,

is also surjective. (3):

Let

P

be a proper prime ideal of

be the canonical map of D = R/P .

Then

R[X,X-1] onto

R

and

D[X,X-1] ,

D[X,X_1] = D[a(u),a(u)~1 ] .

Since

Z

is torsion— free and cancellative, Theorem 11.1 shows that a(u) where

is a trivial unit of d

is a unit of

and this implies that

D .

D[X,X_1] --- say Thus

k = ±1 .

a(u) = dX^ ,

D[X,X"*] = D[X^,X"^] It follows that

u^ e P

, for

307 i

{- 1 ,1} , and hence (3) = >

(2):

ik

is nilpotent.

We assume that

(3)

is satisfied.

u 1 = Z? i= cv.X* i

.

tent and that

R = (u^)

integers

We first establish the inclusion

q .

Theorem 11.3 shows that © (u^)

u

Let

has unit con-

for all sufficiently large

R[X,X *] s R[u,u_1]

under the additional hypothesis that

u^

and

is a unit of

ideal of

R

i * 1 .

Then

R

u ^

is nilpotent.

generated by the cofficients Bu

u = u^X

the case where If

a

and

B = Bu

a(u) = u*X , where

(u|)'1X 1 .

It

each integer

n

.

and

t

unit of r~* . n

R For

over the range

be the

u , where

n

.

of

Bu .

R[X,X"*]

to

k > 2 .

(R/B)[X,X *]

e B[X,X"'I‘]

for each

n .

r Let

n

for

is a s„ = n

u.s.u* = u.X* i i i

v = u -

a £ i £ b , then

> taken over

2 -1 v = u ^X (mod B [X,X ])

and the induction hypothesis implies that To complete the proof for this case potent), we need to show that

,where

v

(u^

R[X,X a unit,

is a unit of

c R[v,v *] . u ^

nil-

R[u,u *] ;

this follows immediately from Theorem 22.2. We observe that the case just considered also handles the case where

u^

is a unit and

u1

is nilpotent, for

the proof shows that under these conditions the cofficient of

X

in

u "1

,

a(u *) =

un = (i^X)11(mod BtX.X-1])

un = r Xn + t n n

Hence if

If

Consider

u* = u^ + B , and hence

follows that Thus

k

has order of nilpotence

a < i < b , i * 1 , w e have

(mod B 2 [X,X *])

of

R[X,X"*] = R[u,u *] .

is the canonical map of

then

u^

Bu

is nilpotent, and to prove the result, we

use induction on the order of nilpotence k = 1 , then

Let

is a unit and all other coefficients are

308 nilpotent. u i s

The remaining case is that where neither

nilpotent.

Let

e^ and e2

sponding to a decomposition u = e^u + e 2u , and where

e^u *

Re^ = (u?)

e^u

in

R .

Re^[e^u,e^u *]

As an element of

e^X

Then

, and

Re1 [e^X,e^X*1 ] ,

e iu i

in e^u is

, and the coefficient of

nilpotent element.

of

R ^ u " * ] = Re^[e^u,e1u *] © R e2 [ e ^ . e ^ 1] ,

e 2u_1 •

the coefficient of

idempotents corre­

R = (u^) © ( u ^ )

is the inverse of

similarly for

be

nor

> a unit of

e ^ X i s

e iu _i > a

Since all other coefficients of

e^u i

are

nilpotent as well, it follows from the case considered above that

Re^fe^Xje^X 1] = Re^[e^uje^u-*]

ment,

.

By a similar argu­

R e 2 [e2X , e 2X *] = R e 2 [e2u,e2u~* ] .

Therefore

RtX.X'1] = Re1 [e1X,e1X' 1] 9 R e ^ e ^ . e ^ ' 1] = Rfu.u'1] ,

and

this completes the proof of Theorem 22.3. In order to pass from

AutRR[Z]

from Section 11 that a set

to

AutRR[Zn ] , recall

E = {e^,...,e^}

of nonzero

lr

orthogonal idempotents of split the unit

u e R[S]

vertible elements t e R[S]

R[ X±gl ,...,X±gn] ring in of

n

conversely, of

if

1 =

is said to

if there exist

u = vCE^e^X^i) + t .

Zn .

We regard

t where

variables over

R[Zn ] , each

with

£i>* ••>£]< 6 ^ » and a

such that

a free basis for

R

Xgi

R[Zn ]

R[Xg l ,...,Xgn] R .

Under an

maps to a unit

u = (u^,u2 ,...,un )

R[Zn ] , there exists a unique

u^

is an

a unit

v e R , in ­

nilpotent element Let

be

as is the polynomial R— endomorphism of

R[Zn ] .

And

n— tuple of units

R— endomorphism

<J>u

309

ofR[Zn ]

such that

u (Xgi)

=

the canonical extension to onR[Xg l ,...,Xgn]

u^ for each

R[Zn ]

i ;

u

is

of the substitution map

that maps Xgi

to

u^

for each

i .

As in (22.1) and (22.3), we are therefore left with the problem of determining conditions on the (u^,...,un )

in order that

u

n— tuple

u =

should be an automorphism.

This is where the notion of splitting comes in; Theorem 11.16 shows that there exists a set splits each

of idempotents of

R

that

u^ .

THEOREM that

E

22.4.Assume that

u = (u, ,...,u ) 1 n

preceding paragraph.

and

R is a unitary ring

d> Tu

and

are as described in the

The following conditions are equivalent.

(1)

u

is an automorphism.

(2)

<j>u

is surjective (that is,R[Zn ]

=

RfuJ1 ,...,!!*1 ]) . (3)

Let

E =

idempotents of splits each

R

u^ .

t^ +

he a set of nonzero orthogonal such that For

Re^

.

and such that

1 £ i < n , write

» where

is a unit of

1 =

t^ e R[S]

Then

E

u^ =

is nilpotent and where

{tKj >1 = 1

a basis °f

u^

f°r

1 < j < m . (4)

There exists a set

E

u^

is as represented there, then

Zn

for

as in

(3)

{bjj ^i=i

such that if each a basis of

1 < j £ m .

Proof.

(1) <= >

Thus assume that

(2): We show that

Z r ...... u*1. ..uln = 0 X1

xn

1

(2) implies .

(1)

.

As in theproof of

n

Theorem 22.3, there exists a Noetherian unitary subring

R^

310 of

R ±

containing each

g

±

r.

.

such that

g

R-^fX

n ] = R-^[u*1 ,...jU*1] .

tion of

<J>u

to the Noetherian ring

hence injective.

Therefore the r e s t r i c ­

R-^[Zn ]

It follows that each

is surjective,

r.

= 0 , so • ,xn

<J>u

is also injective. (2) = >

(3):

1 < j < m , let N

u

•

of

R[Zn ]

Cj If

is a unit of

Let

N

be the ideal of Vjj = u ^

R^ = R/C^

to

be the nilradical of

+ Cj

for each

±b,. ±b . 1J,...,X n:>]

R[Zn ] ; see Theorem 9.9.)

That

Zn , and since

Zn

is a basis for (3)

(4) = >

v^.

b , . ±1 b . ±1 3 ) ,...,tvnjX n j ) ] =

(We have tacitly used the fact that

(bij>i_i

i , then

Rj[Zn ] , we obtain the relation

Rj[X

generates

generated by

, and under the canonical homomorphism

Rj [Z ] = Rj [ O j j X

of

R

R , and for

implies (2):

.

N[Zn ]

is the nilradical

It follows that

{b^

has torsion— free rank

n ,

Zn .

(4)

follows from Theorem 11.16.

Let the notation be as in

(3) .

R[ X±gl,. . . ,X±gn] = E? =1 © R e ^ e i X * 8 !, . . . ,eAX ±gn]

Since

and since

Rtu*1 ,. . . ,u*X ] = E?ssl © Re._te.uJ1 , . . . ^ u * 1] , it suffices to consider the case where loss of generality that is nilpotent and ideal of

R

v^

m = 1 .

Thus, we assume without

ik = t^ + v^X^i

is a unit of

R .

t where

t^ e R[Zn ]

Let

be the

generated by the coefficients of

Bu

t^,t2 ».-->t

•

311 Then

Bu

is a nilpotent ideal of

inclusion

R , and to establish the

R[ X±gl ,... ,X±gn] c R [u^*,...,u**]

duction on the order of nilpotence then

k

of

, we use in­

Bu .

If

k = 1 ,

Rfu*1 ,....u*1] = R[ X±bl,...,X±bn] = R[X±gl ,...,X*gn]

Consider the case where k > 2 .

If

a

a(u^) = v^X^i

aCu^1) = (vt) *X b i .

integers

a^,...,a

a, x

has order of nilpotence

is the canonical map from

(R/B)[Zn ] , then hence

B = Bu

.

, where

R[Zn ] v? =

to + B , and

It follows that for arbitrary

that

a a. a l n. _ •••un E v i •••vn

a,b,+...+a b l l n n / •% d r7^i ^ (mod B [Z J) .

Therefore

O)

X

where

a-,b, +. . .+a b n n =c

c

a, a u1 ...un + f , a 1a9 ...a 1 n a1a 7 ...a 1 it n 1 z n

is a unit of

1 2 **

Next we modify

u^,...,un For

monomials

> where

y^.

Thus,

(*)

w ij € Zb^©...@Zbn .

be expressed in the form

n

to obtain elements

h^,h2 »...,hn , as follows.

R [ u ^ , . . . ,u*n ]

e B[Zn J .

R and f

1 2 *''an

1 < i £ n , t^ e B

is a sum of

and where

implies that

+ P^

, where

is nilpotent, and where

can z^

e

p^. e B2 [Zn ] .

Let

h^ = u^ - 2zij > t^ e sum being taken over all monomials y ^ X w ij hi

in

t^ .

By choice of the elements

c R[u^ ,...,u" ]

where

t* € B 2 [Zn ] .

and

h^

has the form

z^

,

v^X 1 + t* ,

The induction hypothesis implies that

R[Zn ] 9 R[h-^1 , . . . ,h 1 ] , and Theorem 22.2 shows that +1 ±\ R[h^ ,...,hn ] c R[u"

+i

, ...u~ ]

.

matical induction, the equality

By the principle ofmathe­ R[Zn ] = R [ u ”1 ,...,u ~

follows. The automorphism

group of R[G]

, where

sion— free but not finitely generated, but in the case where

R

Theorem 11.6 shows that

G is

tor­

is not known in general

is reduced and indecomposable, R[G]

has only trivial units.

For

any group ring with this property, Theorem 22.5 indicates how the

R— automorphisms of

R[G]

arise.

THEOREM 22.5.

R

a unitary ring with group

units

U(R)

.

Let

Assume

be

that the group ring R[G]

of

admits only

trivial units. (1) UgX^g

If

for

(j) e AutRR[G] some unit

<J>:G — *■ U(R)

, then for each

u^ e R

and some h^ e G .

defined bya(g) = Ug

larly, the mapping

ip:G — ► G

g e G ,

is

a

=

The mapping

homomorphism;

defined by

iKg) = hg

simi­

is

an automorphism. (2)

Conversely,

ip e Aut(G) such that

<J>(Xg ) = a ( g ) X ^ g^ (1):

g,,g, e G , then 1 L

, then

is a homomorphism and

for each g

By definition,

augmentation map on

0

a:G — ► U(R)

, then there exists a unique element

Proof.

<*(81 +

if

R[G]

.

(Xg ) e R , SO

a

u X gl g2

= hgx + hg2 =

e G .

a = y°<J> , where

Hence

g l+g2 if>(X ) = u

y

is the

is a homomorphism. h g l+hg2

+ ^^g 2^ • Xg eR

since

,so

Moreover, if (|>

If

is an


313 R—automorphism. To see that

ij;

Consequently, is surjective,

uXg for some u e U(R) , g e - 1 c u X , and hence iHg) = s . (2): an

Since

{Xg | g e G}

R-module, there exists

g = 0

^

_1

c

Then


It follows

that

<J>(Xg ) =

is a free basis for <J> e

at most one

To verify that

(X )

R[G]

AutRR[G]

the property described --- namely, the map <J>(EagXg ) = EagOi(g)X^g^ .

is injective.

s e G .

take G.

and

=

as

with

defined by

so defined is

a homomorphism, we need only show that it preserves multipli­ cation of basis elements. g-, g 2 (X AX n

= a(gl + g 2)X

Thus, 'Hgn+go) iKgnJ+lKgo) 1 = aCgl)a(g2)X L

gl g? = 4>cx )(j)cx n Since

\{j

is an injection,

EagaCg)X^g^ = 0 0 , for each clear that

.

implies that

g . R

it follows that an equality

Therefore

and

<{> isinjective.

Finally,

{ X ^ g ^ | g e G) = {Xg | g

tained in the image of This establishes

aga(g) = 0 , and hence

<J>

so that (j)

e G}

ag = it

is

are con­

isalso surjective.

(2) .

If the notation and hypothesis are as in the statement of Theorem 22.5, we denote by

Hom(G,U(R))

morphisms from

.

G

into

U(R)

abelian group under pointwise

The set

the set of homo­ Hom(G,U(R))

multiplication -- that

is an is,

(a1a 2)(g) = a 1 (g)a2 (g) ; the identiy element of Hom(G,U(R)) is the trivial homomorphism of

G

a 1 ,a2 e Hom(G,U(R))

€ Aut(G)

^1,^2 then

6 AutRR[G]

, if

into

are defined as in

(2)

U(R)

.

If

, and if of Theorem 22.5,

314 = 4>1 Ca2 (g )x ','2 ( g ) )

= a 2 ( g ) * 1 (X *2 (g ))

.

= a 2 ( g ) a 1 (i(;2 C g ))X ,*'l*-,*'2*-g-*-*

Thus, if

<Jk

tion in

is denoted by

AutRR[G]

(#)

C^1>0^) C^2>a2^ = ^1^2* (ai0^2^a2^ •

isomorphisms of

Aut(G)

AutRR[G]

with their duct of

.

and

shows that

Hom(G,U(R))

and

AutRR[G]

a -- ► (l,a)

are

HornfG,U(R)) , respectively, Aut(G) , then

and

Hom(G,U(R))

AutRR[G]

, and part

n Hom(G,U(R)) = {1} .

is the p r o ­

(1)

of (22.5)

Finally,

is a normal subgroup of

is a split

extension

(#) AutRR[G]

Hom(G,U(R))

, by

; we state the result formally in Theorem 22.6.

THEOREM 22.6.

Assume that

that the group ring be the unit group of homomorphisms of tion, and let

of

(^,1)

Hom(G,U(R))

Aut(G)

Then

and

AutRR[G]

shows that

Aut(G)

ij; — ►

If we identify

images in

Aut(G)

and hence

, then the multiplica­

is given by

It follows that the maps

into

(1/^,0^)

G

R[G]

into

U(R)

Hom(G,U(R))

by

U(R)

be the group of

be the automorphism group of

Aut(G)

Let

under pointwise multiplica­

AutRR[G] - [Horn(G,U(R))]Aut(G) Hom(G,U(R))

is a unitary ring and

has only trivial units.

R , let

Aut(G)

R

G .

, a split extension

.

Section 22 Remarks An by the

R— automorphism n— tuple

case of a field AutRR[X^,...,XR ]

<J>

of

R[X^,...,Xn ]

(f^,...,fn ) , where

is determined

f^ = ^(X^)

.

In the

R , work on the problem of determining has been focused on what is known as the

315 Jacobian Conjecture. automorphism of

To wit, if

— ► f^

determines an

R[X^,...,Xn ] , then it is straightforward to

show that the determinant of the Jacobian matrix J(f^,...,fn ) = [8f^/8Xj] converse fails for

is a nonzero element of

char R * 0 , even for

Jacobian Conjecture states that for dition that X^ — v

f^

det(J(f^,•••,fn )) e R\{0}

to a unit of

R[Zn ]

and

R[X^,...,Xn ] .

that maps each

may induce an automorphism of

Y -- ► X 2Y

For

We note that anon-sur-

R[X^,...,Xn ]

For example, the automorphism of X -- ► XY

n = 2 ; the

implies that

determines an automorphism of

R— endomorphism of

The

char R = 0 , the con­

a nice survey of this topic, see [17]. jective

R .

R[X,Y]

X^

R[Zn ] .

determined by

has this property.

The method used in the proofs of (22.3) and (22.4) can be used to show that, in general, a surjective phism of an affine ring R

is injective.

T = R[t^,...,tn ]

In particular,

each finitely generated monoid that

S

R [S ] S .

R—endomor­

over a unitary ring

has this property for

Without the hypothesis

is finitely generated, the conclusion need not hold;

for example,

X 1 -- ► 0 , X^ -- ►

a surjective

R— endomorphism of the polynomial ring

R [(Xi>1]

that is not injective.

R[[X^,...,Xn ]] jective

for

i > 2

determines

The power series ring

also has the property that each of its su r­

R— endomorphisms is injective

(see [42],

[57]).

For

a general discussion of conditions under which injectivity follows from surjectivity, see [115]. As indicated in the remarks at the end of Section 11, the class of group rings

R [G ]

with only trivial units has

been investigated in the literature, but it seems unreasonable

316 to expect a strong result characterizing such rings for arbi­ trary

R

and

G .

Besides the references on this topic

listed in §11, see also [37— 38], If

R = R 1©...©Rn

units for each

is such that

i , then

direct sum of the groups

AutRR[G] AutR R^[G]

are known from Theorem 22.6. a description of duced and

G

AutRR[G]

R^[G]

has only trivial

is isomorphic to the , and the latter groups

Lantz in [89] has also given in the case where

R

is r e ­

is torsion— free.

Further results concerning

AutRR[G]

, including a gen­

eralization of Theorem 22.6, can be found in [102].

317 §23.

Coefficient Rings in Isomorphic Monoid Rings.

Assume that the semigroup rings isomorphic.

Are

R^

and

R2

R-^ [S ]

isomorphic?

been considered extensively in the case R^[S]

is the polynomial ring in

Even in the case of

n

R 2 [S ]

are

This question has

S = z|j , and hence

indeterminates over

Z q , it is known that

not be isomorphic.

R^

and

R^

R2

need

In fact, an example due to Hochster [77]

exhibits nonisomorphic Noetherian domains that

and

I.

R^[X] - R 2 [X] .

R^

and

R2

such

Work on the isomorphism question for

polynomial rings is too extensive for inclusion here, but references to some work on the question are included in Section 23 Remarks.

Instead, we concentrate in this section

on the case where

R^

on the case where

S

and

R2

are fields, and in Section 24

is a torsion— free group.

This is the only section of the text where results are proved for semigroups that need not be commutative.

Theorem

23.1, Corollary 23.2, and the summarizing result of the section --- Theorem 23.17 ---- do not require that the semi­ groups in question are commutative.

To distinguish these

cases, we use the notation

RS

for the semigroup ring of

over

S

need not be commutative, we

R

in the case where

write the semigroup operation in the elements of

RS

S

as multiplication, and

are written in the form

semigroups considered in results

is used in that part of the section.

shows that if

F

and

K

Si=ir isi •

(23.3) through

commutative, and the more familiar notation Er^X5!

are fields,

S

S

and

contains a periodic element, and

T ^e

(23.16) are

R[S]

and

Theorem 23.17 T

are semi­

groups,

S

FS = KT , then

F - K .

The proof proceeds by reducing the question for

318 general

S

and

T

to the case where

dic commutative semigroups.

S

and

T

are perio­

The first result provides a re­

duction to the case of commutative semigroups.

We continue

to assume that all rings considered are commutative. THEOREM 23.1. is a semigroup.

Assume

R

is a ring with

R = R

There exists a smallest congruence

such that the factor semigroup

S/~

%

Then

I

that the residue class ring Proof.

generates a congruence ~

~

on

on

S I

onto

RS

such

is commutative.

A = {(xy,yx) | x,y e S} c S x S .

Let

to show that

RS/I

S

Let

RS

is the unique minimal ideal of

and ~

is commutative.

be the kernel of the canonical homomorphism from R[S/~]

2

Then

A

S , and it is straightforward

is the least congruence on

S

with commuta­

tive associated factor semigroup. We show that

I

is generated by

B = {rx — ry | r e R , x,y e S , and Let

J

J c I

be the ideal of is clear.

n = 1

generated by

, and use induction on

is impossible, and if [s^

r l [s 1 ] + r 2 ts2 -l = 0 * implying Therefore

f =

- i'1s 2

n > 2 , the relation i * j .

Then

r ^

- r^. eJ

n

hypothesis,

e J , so

Clearly

RS/I

s^

with

[s 1 = 0

has fewer than f^

B .

. The inclusion

To prove the reverse inclusion, take

^i=lr is i e I\(0) case

RS

x ~ y}

and

n = |Supp(f)|

and

is commutative.

.

The

n = 2 , then = [s2 ]

and

~ s 2 and implies

ri = f eJ

s^ ~ Sj

f 1 = (f - (r ^

elements in its support. f e J

f =

~ r2 * .

For

for some

- r ^ )) € I

By the induction

I =J . Let C be

any ideal of

319 RS

such that RS/C

on

S

by

is

commutative.

x y y if and

only if

It is routine to show that S . We verify that

y

y

Define a relation

rx - ry e C

for all

is a congruence on

x y y , z e S , and r e R

rxz - ryz

and

k Z. ,a.b. J=1 3 J For each

rzx - rzy are

where

a.,b. J’ J we have

j

e R

r e R .

is an equivalence relation on S

by showing

that it is compatible with the semigroup operation. assume

y

Thus,

. We must showthat

in

C .

: this is

Write

r =

possible since H

2 R = R

a.b.xz - a . b . y z = (a.x — a.y)b.z e C , and 3

3

3

3

3

3

3

a.b.zx - a . b . z y = a.zfb.x - b . y ) 3

3

3

3

7

3

3

3

e C

.

v rxz - ryz = E^fa^b^xz - a^.b^yz) e C , and similarly,

Hence

rzx - rzy € C . s,t e S

and if

We show that S/y is commutative. If v r = Z^a^b^ e R , then a^b^st - a^.bjts =

a^.s-bjt -bjt*a^.s e C this implies that commutative. x ~ y .

since

rst - rts

Therefore

Then

x y y

so

RS/C e C , so

y > ~ . Let rx - ry

e

B = {rx - ry | r e R , x ~ y}

Since

lows that

is commutative.

C

As above

st y ts

and

x,y e S

be such that

for each

S/y

is

r e R .

generates

I , it fol­

I c C , and this complets the proof of Proposition

23.1. COROLLARY 23.2. R2

are idempotent.

where

~

and

respectively, Then

y

Assume that Let

S' = S/~

R^S = R 2T , where and

let

S

and

T' = T/y ,

are minimal congruences on

S

and

T ,

such that the factor semigroups are commutative.

R ^ ’ = R 2T ’ . If

R^

and

T

are commutative and if

<J>

is an

f 320 isomorphism of isomorphism

R^ [S J

<|>*

where

NCR^S])

R^ [S ]

and

of

onto

R 2 [TJ

* then

R 1 [S]/N(R1 [S])

and

N(R2 [T])

onto

R 2 [T ] , respectively.

N(R1 [S])

istic

.

If

R1

and

R2

are re­

p , Theorem 9.4 shows ~p

of

S , and a similar statement holds for

On the other hand, 0

R 2 [T]/N(R2 [T]) ,

is the kernel ideal of the congruence

p— equivalence on R 2 [T]

induces an

denote the nilradicals of

duced rings of prime characteristic that

<J>

if

, then Corollary 9.12

[S]/N(R^[S]) = R 1 [S/~]

R^

of character­

implies that

, where

asumptotic equivalence on

is a domain

S .

~

is the congruence of

The next result follows imme­

diately from these observations. THEOREM 23.3. semigroups and (1) teristic

If

Assume that

R^S] R^

S

and

T

are commutative

= R 2^T ^ *

and

R 2 are reduced rings of

prime charac­

R 1 [S/~ ] - R 9 [T/~ ] , and each of these

p , then

i

p

p

l

rings is reduced. (2) then

If

R^

and

R 2 are domains of characteristic

R ^ [S/~] = R 2 [T/~]

0 ,

, and each of these rings is reduced.

Our next reduction of the isomorphism problem is to the case of periodic semigroups. periodic semigroups.

We denote by

We denote by

S*

elements of the commutative semigroup

S*

the set of

the set of periodic S .

The hypotheses of

results through Theorem 23.16 either assume or imply that S* *

In particular,

THEOREM 23.4.

Assume

S* * <J> R

an abelian torsion— free group. the ideals

(f)

and

if

S

is a monoid.

is a reduced ring and If

f e R[G]

is such

(f - f ) are idempotent, then

G

is

that f

e R .

321 Proof. u,v e R[G]

Write Let

of

R .

and

f - f2 = v(f - f 2)2

with

g * 0

f = Za Xg . We show that a = 0 for g g by showing that a^ belongs to each proper prime

ideal

P Let

.

f = uf2

be the canonical homomorphism from

integral domain

CR/P)[G] .

R[G]

to the

Then

<J>(f) = <|)(u)(J>(f)2

anc*

♦(f - f2) = ♦(v)*(£ - £2)2 . If

<#>Cf ) = 0 , then

a^ e P .

If

<J>(f) * 0 > then the first

of the equations above implies that

(R/P)[G]

and

If

(j>(f)

then

is a unit of

^(f)

(R/P)[G]

.

(f) = <J>(f2) = <J>(f)2 >

is the identity element of

Supp(<J>(f)) = {0}

and

a^ e P .

is unitary

(R/P)[G]

Finally,

if

; hence

<J>(f)

and

2

<J)(f ~ f ) of

are nonzero, then each is a unit.

(R/P)[G]

But the units

are trivial by Theorem 11.1, so that if

Supp(<J>(f)) = {s} , then

Supp(<J>(f) -(f)2) = (s,2s)

must be a singleton set.

Thus

THEOREM 23.5. that

R[G]

ideals where

(f) G*

and

s = 0 , a^ e P , and

Assume that

is reduced.

If

(f - f2)

Write

Supp(f) u Supp(u) u Supp(v)

mand of H , say

H = H* © W

f , u , v e R[H] = R H * [W]

with

. Since

f e R [G *]

,

as in the

be the subgroup of

H*

such

G .

f = uf2 , f - f 2 = v(f — f2)2 H

e R .

is such that the

are idempotent, then

generated, the torsion subgroup

f

is an abelian group

f e R[G]

proof of Theorem 23.4, and let rated by

G

is the torsion subgroup of

Proof.

, which

. Since of

W R[H*]

H

H

G

gene­

is finitely

is a direct sum­

torsion— free. is reduced,

We have

322 f e R[H*]

by

Theorem 23.4.

The result then follows since

R[H*] c R[G*] . THEOREM 23.6.

Assume that

lative semigroup such that f e R[S]\(0)

S

R[S]

is reduced.

is such that the ideals

are idempotent,

then the set

S*

is nonempty,

is a monoid,

and

S

Proof.

The nonzero ideal

potent element of

Supp(e)

is periodic. w .

R[G]

S

and

is a quotient ring of

(f)

the torsion subgroup of To

S S

G .

S .

(n + l)s = s , which says

since

S

n > m

w

is an

and is therefore reduced.

Thus

thus

with

contains

The group ring

f € R[G*]

, where

f e R[S]

ns = 0

that

for some

s e S*

.

such that

is cancellative.

G*

is

S n G*

=

n R [G *] =

complete the proof, we check

s e S n G* , then

n,m e Z +

S*

is a monoid.

R[S]

If

and

S

is generated by an idem-

is cancellative,

S* .

(n - m)t = 0

of

.

S* * and

From Theorem 23.5 we conclude that

exists

(f - f2)

f € R[S*]

be the quotient group of

R[S n G*] .

and

of periodic elements

Hence

Because

identity element for G

(f)

If

e , and Theorem 10.6 shows that each element

an idempotent

Let

is a commutative cancel­

If

n e Z+

,

and

t e S*

, there

nt = mt , implying Thus

t e S n G*

S n G* = S* . Theorem 23.10 is the last of the series of four propo­

sitions beginning with Theorem 23.4 that have basically the same form.

The hypothesis in (23.10) differs from that in

(23.6) in that we no longer assume that

S

is cancellative.

The proof of Theorem 23.10 uses the Archimedean decomposition S = uS„ a

of

S

introduced in Section 17.

We recall from

323 Theorem 17.10 that each of

S

if

S

S&

is a cancellative subsemigroup

is free of asymptotic torsion.

Two preliminary

results are needed in the proof of (23.10); the first of these concerns the Archimedean decomposition of THEOREM 23.7.

Assume that

S

is a commutative semigroup

that is free of asymptotic torsion. Us

= {t 6 S | s + S c (1)

Us

s

= U, t (2)

b +

is a subsemigroup of Sg

for each

t e S

If a,b

e Ssand

c , then a =

S

s

t 1 ,t2 e

c

e Us

in

aresuch that a + c =

(1) that v

U

6Z+ anc*

and

t l + t 2 + S1

+ s 2 6 tl + t2 + S *

k 2s = t 2 + s2 .

s + S c radft^

+ t 2 + S)

the

U s

+S

s -

sl ,s2 6 Then

U

s

u e U

follows since

Write

k € Z+ and

+v + w

To establish

,v e s ’

. s

We

; the inclusion

rad(u + v +S) c radfv + S) kv = u + w ,where

such that

(k^ + k 2)s =

and t^ + t 2 e U s .

c S , take s ’

S

Therefore

rad(u + v + S) = rad(v + S)

(k + l)v = u

S„ c u and s _ s

is closed under addition, take ’

s

kis = t^ + s^

show that

Moreover,

s

Choose

inclusion

containing the

follow immediately from the definition of

that .

S

b.

t e Sg

. To see

s e S , define

of s as an ideal.

Proof.The assertions ----for

For

rad(t + S)} .

Archimedean component U

S .

u + v + S c v + S .

w e S .

Then

implies v e rad(u + v + S) , and

rad(v + S) crad(u + v +

S)

. This completes the

hence

proof

of

(1) . (2) (1)

: Since

shows that

a + c = b + c , then c + a e S

s ’

, and S

s

a +(c + a) = b +(c is a cancellative

+ a)

.

324 semigroup since fore

S

is free of asymptotic torsion.

There­

a = b . THEOREM 23.8.

group and that

p e Z+

congruence on (1)

Assume that

S

is prime.

is a commutative semi­ Let

~

be the cancellative

S .

If

S

is free of asymptotic torsion, then so

IfS

is

is

S/~ . (2)

Proof.

p— torsion— free, then so is

We recall that

if and only if

[b]

(2)

nb

= [mb]

+ x

m(b

.

and

ma

x e S .

n , m such that

+ y = mb + + x + y)

+ y = b + x

torsion.

9.17 shows that reduced,

is not

since

Consequently,

Assume that

(iii)

+y

(ii)

p

S

where

~

(iii)

and

na

+ x

= =

such that

y . Therefore

m(a +

x+ y)

= n(b + x + y)

, and

a + x + y

Hence

Sis free of asymptotic ~b

and

[a] = [b] . Theorem (i)

R

is

is free of asymptotic torsion, and

p— torsion— free. and

[nb]

is reduced if and only if

is regular on

(ii)

We prove Assume that

is a commutative semigroup.

R[S] S

a

a ~ b

Choose rela­

[na] =

b + x + y are asymptotically equivalent.

a +x

by

is similar, but easier.

There exists x,y € S

+ x + y) , n(a

and

for some

.

S

e S/~ are asymptotically equivalent.

tively prime integers [ma]

is defined on

a + x = b + x

(1) ; the proof of [a],

~

S/~

R

for each prime

p

such that

S

Theorem 23.8 shows that properties

transfer from

R

and

S

is the cancellative congruence on

obtain the following corollary.

to

R S .

and

S/~ ,

Thus, we

325

COROLLARY 23.9.

Let

~

on the commutative semigroup so is

be the cancellative congruence S .

If

R [S ]

is reduced, then

R [S/— ] . Corollary 23.9 is the final preliminary result needed in

the proof of Theorem 23.10, which will prove to be a key r e ­ sult in the ultimate proof of Theorem 23.17. THEOREM

23.10.

group such that

R[S]

that the ideals

(f)

S* *

and

is a commutative semi­

is reduced. If f e R[S]\(0) is such 2 and (f - f ) are idempotent, then

The proof that

in the proof S

.

is the

same as that given

Let S = uSa

be the decompo­

into its Archimedean components, and let S

Since

if and only if

R[S]

is reduced,

totic torsion, and hence each

S&

rad(a + S) = S

is free of asymp­

is cancellative.

the result, we use induction on the cardinality equivalence classes under Supp(f) If

n

m

To prove of the

represented by elements of

. m = 1 , then

f e R [Sa l .

Supp(f) c S&

a e S

for some

and

To verify the theorem in this case, it is suf­

ficient, by Theorem 23.6, to* show that generate idempotent ideals in

f

and

Toward this end, we write

f - f2 = v(f - f 2)3

for some

f , f3 , f - f 2 , and

(f - f 2)3

u = u.+...+u , where x r

f - f2

R [Sa l > f°r then

f e R [S *] c R [S * ] .

Write

n

that gives rise to this decomposi­

tion --- that is, a n b + S)

S* * <J>

of Theorem 23.6.

be the congruence on

rad(b

S

f e R [S* ] .

Proof.

sition of

Assume that

u,v e R [S ] .

f = uf3

The supports of

are all contained in

Supp(u-) c S l a

and

Sa .

and where

326 *S

S ai

for

i* j .

Then

r 3 = E,u.f 11

f

aj

Supp(u.f^) s S . i a^+a

It follows that

sum being taken over those over the same indices 2

f =(Zu. f)f ideal of

.

a

such that

Therefore

f

, the

j

Sa +a = Sa . Summing aj Eu.f e R[S ] and j a

generates an

idempotent

2

R [Sa ] •

By the same argument,

rates an idempotent ideal of the case

f =

j , we note that

2

e R[S]f

J

j

, where

f - f

also gene­

R[Sa ] , and this establishes

m = 1 .

We assume that the result is true for

m < n , where

n > 1 , and consider the case where the elements of determine tains b +

n

classes under

an aperiodic element S £ r a d ( s + S)

there exists taining

s

R[P]

b

R[S]

is an ideal and

.

s e Supp(f)

Consider

; here of

f - f

idempotent ideals of

the

S\P

.

If not, then

2

.

Let

Then

, so

R[S\P]

P

of

n

implies that

f^ e R [S *] ,

b e Supp(f^)\S* asserted. ment of

since

f^

.

is the

and

^

- f^

R[S]

.

The number of

It follows that Supp(f)

that b

S^contains

and thatSupp(f)

{t e S | b + S c rad(t + S ) } .

generate

Supp(f^)

The induction hypothesis

and this is

We conclude

S

f = f^ + f 2 ,where

fj -

and of

Sg c P .

con­

decomposition R [S ] =

Archimedean components represented by elements of is less than

S

is a subsemigroup of

R [S ] .

f 2 e R[P]

R[S\P]— component of

Supp(f) con­

b . We first show that

for each

but not

f 1 € R[S\P]

Assume that

s e Supp(f) and a prime ideal

R[S\P] + R[P] of and

n .

Supp(f)

c

Define

a contradiction,

for

+ S c rad(s + S)

, as

each aperiodic

ele­

=

327 We show that the assumption that tradiction. where

Write

onto

S

S/~ .

Moreover,

and let Theorem

<J)(f)

ar*d

idempotent ideals of I = , then

[b] e

23.6, or else s € Supp(f^)

.

Let

f2 e

~

anc*

be the cancellative

be the natural homomorphism of

23.8 shows thatR[S/~]

is reduced.

2

<|>(f - f ) = 4>(f) ~ <J>Cf) R[S/~]

.

generate

Consider [b] e S/~ .

Since

for c e Supp(f2) , c * b .

Supp(<J>(f))

[b]

and [b] e (S/~)*

{ Supp(<J>(f))

and

There­

by Theorem

[b] = [s]

for some

[b] = [s] e (S/~)*

In the latter case,

s e S* .

since

4>

[b] * [c]

fore either

is empty leads to a con­

f = f^ + f2 , where

Supp(f^) <= (S\S^) n S* .

congruence on S

I

Thus, in either case there exist

m,neZ+ , m * n ,

and

x e S

such that

mb + x

= nb + x .

But this contradicts the assumption that

I =
and

We conclude that

nb

are distinct elements of

I * .

Thus

I

is an ideal of

Theorem 23.7 shows that P

of

S

T = S\P that

containing .

As above, 2 and f - f

f

We note, moreover, that

I n

S , and part = <J> .

Isuch that R[T]

(2)

mb

of

Choose a prime ideal

P n

= <J> , and

is reduced and

let

f e R[T]

generate idempotent ideals of

that the definitions of

I

and

is such R[T]

T

.

imply

J =

J={x

e T 11 ^ + x = t2 + x

for some

By considering the homomorphic where

S^ .

since

~

is the cancellative

t^ ,t2 e S^ , t^ * t 2}

image of

f

congruence on

in R[T/~] T

, we obtain,

as above, a contradiction to the assumption that aperiodic.

Hence

Theorem 23.10.

,

b

is

b e S* , and this completes the proof of

.

328

Theorem 23.10 is related to the isomorphism problem b e ­ cause it can be used to show that if regular ring and S* *

.

S

R

is a (von Neumann)

is a commutative semigroup such that

R[S*]

is the maximal regular subring of

This result is the content of Theorem 23.12; the

proof of that result uses Theorem 23.11. THEOREM 23.11. (1) a ring,

If

is an ideal of

If

S

is a regular ring.

R , then

A , considered as

is a periodic commutative semigroup and if

is reduced, then Proof.

some

R

is regular.

(2) R[S]

A

Assume that

(1):

r e R .

R[S]

If

Since

is a regular ring.

a e A , then

a = ra

3

= ra«a

ra e A , it follows that

A

2

for

is regular

as a ring. To prove unitary.

(2) , we consider first the case where

Let

S^ = S

u

{0}

R[S^]

dim R[S^]

= dim R - 0 .

R[S] g e

R[S^]

and

R[S^] Part where

R[S^]

and where

R[S]

r

S^

is periodic,

It follows that = R + R[S] R

n

, where

R[S] = (0)

g = r + t , where is nilpotent,

is reduced, we then have

.

If

r e R

and hence t = 0 , so

is a zero— dimensional reduced ring, hence regular. (1) R

E^riXsi e I

We have R[S^]

The element

Because

Then

R .

isnilpotent, we write

t e R[S] .

r = 0 .

S .

isintegral over

is an ideal of

is

bethe monoid obtained by ad­

joining an identity element to and hence

R

then shows that is unitary. R[S]\(0) .

be the ideal of

R

R[S]

is also regular in the case

In the general case, take We wish to show that generated by

f =

(f) = (f2) . •

Since

R

Let is

329 regular,

I

is principal, generated by an idempotent

As a ring,

I

is unitary and regular; moreover,

By the case just considered, an idempotent ideal of fore

R[S]

I [S ] , and hence of

Assume

commutative semigroup with W = {f e R[S] | f

of

R [S ]} .

Then

W

regular subring of Proof. clear that hence

and

R

= R[S*] and

R[S]

W

W

by Theorem 23.11.

Assume

R-^ [S ] - R 2 [T]

R x [S*] - R 2 [T*] Proof.

that phism

R1

.

R 2 [T]

Theorem 23.12 shows that

R 2 [T]

(f)

W £ R [S * ] .

Therefore R[S]

and

Then

It is

R [S ] , and W = R[S*]

R2

are regular rings, S* * <J> , R^[S]

T* *

and

R [S ]

R ^ [S ]

is.

contains a nonzero element 2

(f - f )

are idempotent, and hence Theorem 23.10 then shows

By Theorem 23.12, it follows that the isomor­

R^tS] = R 2 [T]

and

K[T]

is

and

induces an isomorphism

R^[S*]

- R 2 [T*].

We show presently (Theorem 23.16) that isomorphism of F[S]

is

.

is reduced since

also has such an element. T* * <j> .

is a

.

The ring

such that

S

.

are commutative semigroups with

reduced, and

f

There­

is the unique maximal

contains each regular subring of

COROLLARY 23.13. T

R [S ] .

S* * <J> , and R[S] is reduced. 2 f - f generate idempotent ideals

Theorem 23.10 shows that

W 2 R [S *]

and

generates

is a regular ring,

the unique maximal regular subring of

S

f

f e I [S ] .

is regular.

THEOREM 23.12.

Let

it follows that

e .

implies isomorphism of

periodic commutative semigroups

S

and

F T

and .

K

for

To handle the

330 case where

S

is not a monoid, we need one preliminary-

ring— theoretic result. THEOREM 23.14. let

P

Let A be an ideal

be a proper prime ideal of

(1)

P is an ideal

for some prime ideal (2)

If

of

Q

R

such that Proof.

rx c A

implies

r e R

rxP c P .

P

is prime in

hence

P

is an ideal of R .

containing

P

i .Hence and

x e A

, it follows that

Q (2):

primes of Then

R

Q

with

and

m < n

A/Pm - (A + Q )/Q m Mn nil

Pm + i = Qm + i n ^ taining

maximal ideal of R

ideals of (2)

{M^ n M} M If

*

(2)

Assume that

R .

distinct from (1)

A/Pm .

f°r some prime

THEOREM 23.15.

of

A\P

rP c A .

Q

in

R .

Let

Q, <...< Q ^1 ^m

n A = P^

is an ideal of

Qm . By induction,

and

of

The

Q n A = P .

and

is a proper prime ideal of

Then

is maximal with respect to

Assume that there exists a chain R

.

of

> dim A .

There exists an ideal

such that

is prime in

A ,

rP c P , and

failure to meet the multiplicative system ideal

dim R

x e A\P

Moreover,

Since

R

A

n

<...< Q r of proper primes

Choose

A

as a ring.

a chain of proper primes of

n A for each

(1) :

and

R .

P^ <•••<'Pn

P^ =

A , considered

R

and is of the form Q

of

then there exists a chain R

of the ring

By

Qm + ^

for

R/Qm ^m

(1) °f

of

1 < i < m . and

Pra,-./P m+l' m

we conclude that R

properly con­

then follows. R

is unitary and

M

is a

be the set of maximal ideals

M , and consider

M

as a ring.

is a set of (distinct) maximal prime

; moreover, M/(M n M ) = R/M for 9 a a

dim R = 0 , then

{M^ n M}

each

is the set of

a

.

331 maximal prime ideals of Proof. so each

(1):

M n M

M .

We have

M/(M n Ma ) - (M + M a )/Ma = R/Ma ,

is a maximal prime ideal of ^

a

ideals are distinct since (2):

If

n M <£ M 0 $

a

for

Pis a maximal prime of

23.14 shows that and hence

M

P

M .

These

a * 6 .

M , then

Theorem

is contracted from a prime ideal of

P = M n M a

for some a

R

,

since

R

is zero—dimen-

and

K

are fields and

sional. THEOREM 23.16. that

S

and

and if

S

T

Assume that

are commutative semigroups.

is periodic, then

Proof.

Let

T is

= S u {0}

adjoining an identity element is possible, even if F + F[S] S^

(2)

S

dim

If

0

to

S ; this construction Then

F[S°]

F[S^] = 0 , and hence dim Therefore

is a homomorphic image of

F[S]

, with the possible exception of

F[S°]/F[S]

= F .

due fields of

F[S^]

are the same as

is a residue field of

K[T°] .

is periodic. Because

dim K[T] = 0

and that

prime

K[T]

P

of

.

since

F[S]

the

, and

.

the same as those of T

by

are the same as the resi­

We wish to show that the residue fields of

that

^ 0

Theorem 23.15 shows

F[S^]

F[S]

Since

dim F[S] = 0

fields of

hence the residue fields of

F[S^] =

F[S]

fields of

F

F[S]

.

.

that the residue

But

F = K .

be the monoid obtained by-

is already a monoid.

of Theorem 23.14.

F[S] - K[T]

also periodic and

F [S ] is a maximal ideal of

is periodic,

part F

, so

F

To do so, it suffices

K[T] - F[S]

K[T]/P

K[T]

are to show

, it follows that

is a field for each maximal

We show that this implies

dim K[T^] = 0.

residue

332 Suppose not, and let K[T^]

.

Since

A K[T]

Q-^

< Q2

be a chain of

is not maximal in

Q 2 n K [T] = K[T] , which implies is a maximal prime of

K[T]

for the nonzero proper ideal

Q 2 = K[T]

Therefore

.

It follows that

, and this is a contradiction,

KfTj/Q^

of the domain

does not have an identity element as a ring.

conclude that

of

K[T^] , then

n K [T ] < Q 2 n K [T ] .

, and hence

KfT^J/Q^

proper primes

dim K[T°] = 0 , which implies that

T°

We is

periodic by Theorem 17.3. We examine the residue fields of maximal ideal of

F[S^]

rally imbedded in periodic,

and u

F n

, then

= F[S^]/Ma

is generated over

fying an equation of the form n > m .

Then

If

= <J> ,so

is

.Moreover, since F

by elements

S^ is usatis­

= 0 , so either

Therefore

a

F is natu­

un - um = 0 , where

um (un-m - 1)

is a root of unity.

F[S^] .

n,m e Z + u

= 0 or

is generated over

F

by roots of unity. Let 0

<J> be an isomorphism of

induces a bisection

of

F[S] Aand a

some

a

F

K[T]

K

K .

K

K

of roots of unity.

lows

u^

is

that

n^

.

a(F)

^ a ^aeA

in such a way

Since F - A

is imbeddable in

Moreover,

accomplished by isomorphisms such a way that

Then

a

for

is imbeddable in each residue field

, it follows that

is imbeddable in

of

K[T]


and since

onto K [T ] .

of the residue fields

onto the residue fields of

that

of

<J>*

F[S]

F . Similarly,

these imbeddings may be

a:F --- ► K

is generated over

and cr(F)

ip:K --- ► F by a set

in {u^}

Assume that the (multiplicative) order Since

iKu^) e F

contains the rrth

has

order n^ , it fol­

roots of

unity.

333 Consequently, fore

u^ e a(F)

for each

i

and

a(F) = K .

There­

F - K . Unlike the results in this section since Theorem 23.3,

the semigroups in the statement of Theorem 23.17 are not assumed to be commutative. THEOREM 23.17. T

are semigroups,

FS a KT .

Then

Proof. isomorphism where S^

S^

Assume S

F and

fields,

S and

contains a periodic element,

and

F = K .

Corollary 23.2 and Theorem 23.3 show that the FS = KT

and

T^

induces an isomorphism are commutative,

is a homomorphic image of

23.13

K are

then shows that

T^ * <j>

F[S^]

F[S^]

is reduced, and

S .

Hence

and

F [S J ] - K[TJ]

Theorem 23.16, we conclude that

= K [T ^ ] ,

Sj ^

Corollary .

Applying

F - K .

Beyond Theorem 23.3, the material in this section appears to have slight application to the isomorphism question in the case of a commutative semigroup

S

such that

S* = .

The

next result resolves what is probably the most natural sub­ case of this problem. THEOREM 23.18. and Then

T

F and

are subsemigroups of

Z+

K are fields and

such that

S

F[S] - K[T]

.

F - K . Proof.

and

Assume

Let

m

and

T , respectively.

n

Define

U = {f e F [S] | for each more than

be the smallest elements of

k

k e Z + , f^ elements of

is not a product of F[S]}

.

S

334 We show first that order

m .

f e U .

If

U

is the set of elements of

fe F[S]

has order

that

rd e S

for each

d

k eZ +

Forthis

, and

the set of elements of the set V =

V

more than

above shows

that

<j>(aXm )

the form

<j)(a)

has

of order


is

for

characterized as

is

is nota product of

F[S]

onto

. K[T]

Let a e F ,

a

bXn + q for some b

€

> n .

we show that the mapping

a^,a2 e F

n

rd

{ U. Similarly,

k elements ofK[T]}

We define

is the coefficient of

u ^
g

of order

k e Z+ , f^

<J>be an isomorphism of

q e K[T]

is of the form

, where

(fe K[T] | for each

Let

F[S]

K [T ]

e F[S] has order

k , we have g^ = (Xm )^+1h ,

some

h e

such

such that

h

Hence

choose M e Z +

g

where the degree of each term in r > M .

be the greatest c om­

and

r > M . If

w > m , then there exists kw - (k + l)m > Md .

S

of

m , it is clear that

To prove the converse, let

mon divisor of the elements of

F[S]

Xn

in

t:F -- ► K

an isomorphism of

F

.

Then the * 0 .

K\{0}

--- ► K <|>(aXm ) •

If

defined by

onto

K

Then

and some as follows:


.

= Suppose

and

(a1Xm ) = b 1Xn + q x , < K a 2Xm ) = b 2Xn + q 2 ,

4>(Xm ) = u X n + q

Then

(jiffa-^ + a 2)Xm ) = (bj^+

oCaj) + a(a2)

and

+

+

^2

’ so ° ^al + a 2^ =

+ a 2) = t U j ) + x(a2) .

Moreover,

335 *Ca1a 2X 3m) = i(i(a1a 2Xln) CXm )= ( b ^ 11 + q ^ (b2Xn + q 2) (uXn + q) =

2

Therefore u

~1

b^ b2uX 3n + (higher degree terms).

u a(a^a2) =

a(a^a2) = u

-

2

= ua(a^)cr(a2)

a(a^)a(a2) = T(a1)x(a2) .

injective homomorphism

from F

is surjective, take

b e K\(0)

where

Then

ord(p) > m .

ord(q) > m .

If

(c - ub)Xn + q

and

into

*r(a^a2) =

Therefore

K.

and let
has order

*(ubXn ) = aXm + p

(cXn + q) - ubXn =

n , whereas

+ q) - <J> *(ubXn ) = aXm - (aXm + p) = - p

greater

than m , a contradiction.

under the mapping

t

.

t

<J>(aXm ) = cXn + q , where

c * ub , then

b = x(a)

is an

To prove that

u Ac , and

t

Thus F

Therefore

and

has order

c = ub , b =

K are isomorphic

.

We remark that not only are the fields

F

and

K

in

the statement of Theorem 23.18 isomorphic, but the semigroups S

and

T

are also isomorphic in that case. Section 23 Remarks

The unitary ring R[X] - S[Y]

implies

R

is said to be invariant if

R - S , and

isomorphism of the polynomial rings invariance of

R

and

S .

R

is R^n ^

n— invariant if and

S ^

implies

Coleman and Enochs initiated the

study of invariant rings in [35]; see [1] and [24] for other results.

The class of invariant rings includes the classes

,

336 of Artinian rings, von Neumann regular rings, and quasi— local domains;

the class is closed under finite direct sum, and it

contains the ring of all algebraic integers in an arbitrary finite algebraic number field.

The analogous problem for

power series rings is considered in [114],

[84], and [68],

where the term power— invariant is used to describe a ring such that ular,

R [[X]] = S[[Y]]

implies that

R =* S .

R

In partic­

[68] contains an example of a Noetherian ring

R

that

is not power— invariant. The case of Theorem 23.17 where

S

and

was proved in [2] by Adjaero and Spiegel. cases of the isomorphism

R-^[G] = R 2 [G ]

[2] besides the case where I K

R^

and

R2

T

Numerous other are considered in are fields.

am unaware of an example that shows for fields

that isomorphism of the semigroup rings

not imply isomorphism of

F

and

K .

FS

F - K .

an isomorphism a field

F

and

KT

F

does

Even if such an ex ­

ample exists, it's possible that isomorphism of implies

are groups

FS

and

KS

Examples mentioned in Section 25 show that

F[G^] = F[G2]

for g.roups

G^

and

G2

and

need not imply isomorphism of

G^

and

G2 .

and

337 §24.

Coefficient Rings in Isomorphic Monoid Rings. II.

In considering the isomorphism

R-^ [S ] = R 2 [S ]

in

Section 23, we imposed conditions on the coefficient rings R^

and

monoid

R^ . S

In this section, conditions are imposed on the

instead.

Most frequently

S

finitely generated torsion— free group. that

R^[Zn ] - R 2 [Zn ]

for some

n

is taken to be a Corollary 24.9 shows

if and only if

R^[Z] = R 2 TZ] , and hence the primary emphasis is on the case where

S = Z .

Another reason for the stress on this case is

that Sehgal poses as Problem 27 of [125, p. 230] the question of whether isomorphism of morphism of and

R2 .

R^

and

R^[Z]

R2

and

implies iso­

for Noetherian unitary rings

R^

In Example 24.7 we present an example due to

Krempa [86] which shows that the question has a negative answer.

All rings considered in this section are assumed to

be unitary.

Before beginning, a word of caution concerning

notation is in order. over

R

Both polynomial rings and group rings

are considered in the section, and brackets are

used to denote each.

But in combination,

consideration is always sistently used to denote

Z , whereas

denotes the group ring of

ring in

X

over

ring in

X

and

R , while

THEOREM 24.1. is an isomorphism of then

and

indeterminates.

R[X][Z]

Y

X

the group under

Z

R[Z][X,Y]

S

R[S]

be either onto

T[S]

are con­

For example,

over the polynomial means the polynomial

over the group ring of Let

Y

z”

Z over or Zn .

such that

R . If

<J>(R) £ T >

R - T . Proof.

Let

be a free basis for

S

<|>

and let

338 =
for each i .

In the case of

T [S ] = T [{11 = <|>CR [S ]) c T[uj,...,un ] s T[S] = T[u^,...,un ] . of T[X

T[S] ±gl

If

T [S ], and hence

S = Zn , then each

is a unit

and as in the preceding sentence,

, ...,X

±gn

the remarks

+1

] = T[u~

+1

,...,u" ] .

By Theorem 22.4 or by

at the end of Section 22, it follows in either

case that there exists that

,we have

ofi^) = Xgi

a

T— automorphism

for each

i .

Thus

a

of T[S]

such

a<|>:R[S] --- T[S]

is an isomorphism that maps the augmentation ideal I J=

= ({1 - Xgi }^) (U

of

- X gi}")

R[S]

of

onto the augmentation ideal

T[S]

.

Consequently,

R * R [S ]/I * T[S]/J = T . If of

G

and

R[G]

H

are groups

ontoT[H]

for each

, then

g e G , <|>(Xg)

is a unit of

T

and

and if



is of

<J)

is an isomorphism

is said to be elementary if, the form

t e T[H]

uX*1 + t , where

is nilpotent.

u

Theorem 24.2

permits a frequent reduction of questions concerning isomor­ phisms

<J>

of

R[G]

onto

T [H]

to the case where


is

elementary. THEOREM 24.2. groups, that

G

isomorphism of sitions

Assume that

R[G]

onto

T[H]

and

.

H

are torsion-free

T = T,©...©T I n

Since

R^[G]

subgroup of the unit group of

T[H]

is an

into nonzero

i ,
onto

(
<J>

Then there exist decompo­

such that, for each

elementary isomorphism of Proof.

and

is finitely generated, and that

R = R.,©...©R I n

subrings

G

T^[H]

restricts to an .

is a finitely generated , Corollary 11.18 implies

339 that there exists a decomposition

1 =

of 1

nonzero orthogonal idempotents such that <J>(Xg) •

ollary 10.8

shows that each

and let

= Tf

.

,and

by definition of the statement

for each each

i

<J>(Xg )

R^[G]

Then

onto

T [Z ]

and

R[Z] If

Let

R^

that

Cor­

=

Re^ = T^H]

F

splits

<j) to

<J>

T[Z]

= uXr

(2)

There exists an

If

Proof.

(1):

a(X) = X .

If

where

R

R

isa unit of

is an ele­

r . * T.

v

of

a of

R[Z]

such

R .

= T.

r = 1 , then Theorem 22.3 implies that a

of

T [Z ]

Considering the isomorphism

Let

<J>(X) = uXer +

t ,where

is nilpotent.

Let o ^

c^fX) =

be such that

such that a<J>of

(2):

a 2 e AutRR[Z]

R[Z]

iscalled the

R— automorphism

T— automorphism

such that

TtX.X'1] ,

follows as in the proof ofTheorem

R[Z]

u

|r|

, it

<{>~A (t) e R[Z]

and

<j>:R[Z] — -w T[Z]

for some unit

r = 0 , then

there exists a

,

deg

Assume that

r = 1 , then

a(X) = vXr (3)

+t

and is denoted by

If

RtX.X'1]

is nilpotent, then

mentary isomorphism of degree (1)

as

is an elementary isomorphism of

<J>D O

THEOREM 24.3.

of

R .

4>(Ri [G]) = 0 (eiR [G ]) =fj[T [H]

and



and if

t € T[Z]

degree of

T[Z]

is in

i .

is elementary.

respectively.

that

for each

,it follows that the restriction of

We regard

T

e^ = <J> 1 (f^) e^

into

F = {f^ ,f 2 »•••fR }

splits each

Ti

Let

e T

24.1

e

= ±1

bethe

R[Z]

onto

that

R = T.

.

Then

R— automorphism

X - *(t)and let a 2 (X)= X G

$(X - <(>-1 (t)) = <J>CX) - t = u X er .

.

Then <J>a^(X)

Therefore

=

340 (Jxj ^c^CX) = <j)Oi(XG) = ueX r . this establishes (3): unit of tions

(2)

By part T .

(2) , we can assume that

and

T =

Tf^[Z]

over, the restriction f^v , a unit of

© Tf^

and such that

isomorphism of

show that

and

a = a

4>(X) = v

onto <|k

such that

and

restricts to an elementary

Re^[Z]

of

<J>

for each

to

for

each i , so

Re. = Tf.

for

each i , it follows

l

i

i .

Re^ [Z ]

Tf^

p e R[Z]

maps

Then

and part

(1)

implies that R^

and

can be imbedded in the other. and

T[Z]

Therefore

R = T

R

and

T

R -T

<J> 1

.

is ele-

is a unit of

R

that

T

and

4>ohas degree

1 ,

. R2

are subismorphic if each

While isomorphism of

need not imply isomorphism

does imply that

If we

(j)((uXr + p)X1"r - p X 1-r) =

u 1 r is a unit of

is nilpotent.

Recall that rings

u

to

e AutRR[Z]be such

a

(a(X)) = <J>(uX) =

u 1 rX - (HpX1 r ) , where (pX1 r) e T[Z]

Let

e^X

that

*

is nilpotent.

More­

deg <Jk = 0 .

—1 T (X) = uX + p , where

Write

a (X) = uX .

is a

^(e^) = f^

Thus, we assume without loss of generality that mentary.

2 ,

.

© Re^ i

v = ue

Theorem 24.2 shows that there exist decomposi­

R =

for each

Letting

of

R

and

are subisomorphic.

R [Z ] T , it

A proof of

this statement uses the following result. THEOREM 24.4. the ring a unit of

where

R , let

Let r

X

and

Y

be indeterminates over

be a positive integer, and let

u

be

R .

(1)

R[X,X~ ^ ,Y]/ (Yr - X) = R[Z]

(2)

RtX.X'-’-.Yj/fY1' - u) = R[y] [Z ]

{l,y,...,y

r *1

. , where

yr = u

and

} is a free module basis for R[y] over R.

341 Proof. (1): x = X + I . {y1 }”^

Let

I = (Yr - X) , y = Y + I , and let

We show that

is free over

R[X,X 1 ,Y ]/I = R[y,y *] , where

R .

R[X,X *,Y]/I = R[x,x \y ]

We have

y r = x , a unit of

. Moreover,

x A = (y *)r , so

—X

i ® x,x € R[y,y ] . To show that {y }_oo is free over v it suffices to show that r^ + r^Y+.-.+r^Y belongs to only if each

r^ = 0 .

R , I

Thus, write

E q I^Y1 = (Yr - X)f(X,Y)/Xm , where

f(X,Y) e R[X,Y]

and

m > 0 .

Substituting

X = Yr

k yields that

^oriY

R[y,y 1 ] = R[Z] (2):

let

= 0 * an(* ^ence each

As in

x = X+ J

and to prove

.

(1) , let

that {xxy^

R .

Thus, assume

It is clearthat

|i e Z q

, y =

that

Y + J , and

R[X,X_1,Y]/J “ Rfy.x.x'1] ,

R .

and

{x1y-^ | i e Z

0 < j < r - l }

r - 1

is free over are polyno­

such that

+ £X (Y)X+...+fs (Y)XS eCYr - u) , say

Z®fi (Y)X1 = CYr - u)h(X,Y)/Xk , so that E®£i (Y)Xk+i = (Yr - u)h(X,Y) Considering divides each

as

R[Y][X]

, it follows that

f^(Y)

in

R[Y]

Since

each nonzero multiple of

Yr - u

By choice of the polynomials f^(Y) = o

.

R[X,Y]

for each

i .

.

and

In fact, it is enough to

fQ (Y),...,fg (Y) e R [Y ]

of degree at most f0 (Y)

J = (Yr - u)

is free over

show

This implies

, as asserted.

(2) , it suffices to prove that

0 < j < r - 1}

mials

r^ = 0 .

Yr - u

Yr - u

is monic,

has degree at least

f^(Y)

r .

, it then follows that

This completes the proof.

34 2 THEOREM 24.5.

Assume that

mentary isomorphism of degree ring

<J>:R[Z] -- T[Z] r .

is an ele­

There exists an extension

T[v]

of T such that vr is a unit of r -1 T,{l,v,...,v } is a free basis for T[v] over

T , and

R - T [v] . Proof.

Without loss of generality we assume that

<J>(X) = uXr , where

u

is a unit of

<J>*:R[Z][Y] -- ► T[Z][Y] ring induced by

T .

Let

be the isomorphism of the polynomial

--- that is, *(Y) = Y

Let

I

be the ideal of

let

J = <(>*(1) = (Yr - uXr ) = C(X'1Y)r -u)

class rings

R[Z][Y]/I

an isomorphism

a

and

for

over

T .

R * T[v]

COROLLARY 24.6. then

R

subring

and R^

such that T

where

T of

The residue

Moreover, the proof of

v

also shows that T[Z][Y]/J v= X ^ Y + J , x = X + J , r —1

}

= vr =

is a free basis

a (y) = *(Y) + J = Y + J =

is a unit of

T[v]

1 , and part

(1)

.

Consequently,

of Theorem 24.3

. If

R[Z]

and

T[Z]

are subisomorphic. T

and

are isomorphic under

{l,v,...,v

We have

is elementary of degree

shows that

.

- X,

R[Z][Y]/I = R[y,y *] s R [Z ] , where

T , and

XX *Y + J = vx , where a

<j>* .

The proof of (24.4)

is a unit of T[v]

T[Z][Y]/J

induced by

T [v] [x,x_1] * T [v] [Z ] , u + J

IR [Z ] = ^ *

R[Z][Y] generated by Y r

Theorem 24.4 shows that y = Y + I .

and

are isomorphic,

In fact, there exists a

and an injective endomorphism

a(T) c R^ c T , where

R^ = R

and where

are finitely generated free modules over

a(T)

a

of R^

and

T and R^ ,

respectively. Proof.

It suffices to show that

T

can be imbedded in

343 R

by means of an isomorphism

generated free module over an isomorphism, T=*T1@...©Tn i

sion ring

of

R and

T^[v^]

<Jk

T

(T) .

Let

T^[v^]

R

<J>:R[Z] -- >-* T [Z ]

be

R = R ^@ ..,0Rn

and

, the restriction of

over T^ .

such that {1,v ^ , ...,vTi

Let

[Z ] for each


to

R^ [Z ] , is

shows that there exists an exten­

°f

R^ , and

is a finitely

such that <J>(R^ [Z]) =

Theorem 24.5

is a unit of

such that

and choose decompositions

and such that

elementary.

t

x

R^ = T ifv iJ » v i*

= ui

is a free basis for

W = T^[v^]©...@Tn [vn ] , let v =

(v^,...,v )

, and let

bedding

of T

in

{l,v,...,vr

*} is

a free basis for W

a:W -- >-* R

is the natural isomorphism, then t = ay:T ----►

y

r = sup{r^}^ . W , we have

Under the natural im­

W = y(T)[v] over

, where

y(T)

.

If R

has the desired properties. We next present an example of non— isomorphic two-dimen­ sional Noetherian domains T[Z]

are isomorphic; EXAMPLE 24.7.

the Let

R

and

T

such that

a = (1 + / = T 7 ) / 2

, let K = Q(a)

D = Z[a]

Then

is a Dedekind domain with class number

be the integral closure of

is the nontrivial automorphism of

K , then

unique nontrivial automorphism of

D

<j>

Z

be the maximal ideal

show that that of fore

I

.

is not principal,

I5 = bD , where D , we have I2

(2,a)

is not equivalent to


induces the

and it also induces an D .

Let

It is straightforward to

that

b = 4 + a .


,

in K . 5 . If

automorphism of the group of fractional ideals of I

and

example is taken from [86].

and let D

R[ Z ]

I n <J>(I) = 2D , and Thus, in the class group

and 1 C I D

<J>(I *) = I . > where

There­

i = 1,2

and

344 j = ±1 .

Let

R =

I2nXn

subrings of the group ring sequences D .

{I2n}°° n=-°°

Since

1,1

D-modules, over

-1

,I

-

, and

I R

Noetherian.

sional Noetherian domain dimension at most

2 .

2

be the Rees corresponding to the

{In }°° n=-°°

it follows that

D , hence

T =

D[X,X *]

and 2

and

of fractional ideals of are finitely generated

and

T

are affine domains

As overrings of the two-dimen­

D[X] , the domains On the other hand,

two-dimensional quotient ring of both

R

R

and

T have

D[X,X *] and

is

T , so

a

dim R =

dim T = 2 . Assume that

R

and T

are isomorphic

Under localization at

Z \ {0> , a

K[X,X

a(K) = K , a|K = <J>*

and

.

a(X) = cXJ , where

j = 1 a

Therefore

or

- 1 .

c a

for

i = 1

I2X

onto

or 2 ,

K

and

I^X-* , since

is surjective and since it maps monomials of degree

a(I2)-cXj = ^ C l ^ c x j implies that

.

and

(Y)

= bX 5Y 2 .

are isomorphic.

t

1

t(R[Y,Y 1 ]) c T [ Y, Y_1]

T [ Y , Y -1] . {I2nXnY^ {In XnY-®

Since

| n,j e l ) | n,j e Z}

Hence

t

Let

R

and

t t

be the

(X) = X 2Y , 1 (X) =

We observe that

and that

t'1 (T[Y,Y'1]) c RtY.Y"1] ;

is an isomorphism of

R[Y,Y_1]

t

determined by

is determined by

b " 1x'2Y , t " 1 (Y) = b 2X 5Y ' 2 .

this will imply that

a(I X) =

, which

, a contradiction.

KfX*1 ^ * 1]

Then

On the other hand, (f>i (I2)=

n

2

We show that the group rings

T[Y,Y *]

K— automorphism of

.

Therefore

I2 e <^(1^)

are not isomorphic.

R[Y,Y *]

t

maps

nj

a .

induces an automorphism of

is a nonzero element of

Clearly

onto monomials of degree

T

under the map

R[Y,Y

is generated by

and since

T [ Y KY *]

is generated by

, it suffices to show that

onto

345 f(I2nXIV ) Since

£ T [Y.Y"1]

and that

t '1 (Ii1X iV

)

£ R f Y . Y 1] .

b e 1^ , we have T(I2nXnYJ’) = I2 n CX2nYnb^X5^Y2^) = j2n^j^2n+5jYn+2j c j2n+5jx 2n+5jYn+2j

which is in

T[Y,Y_1] . T'1 (InX1V )

Similarly, = Inb"nX' 2nYnb 2-*X5^ Y 2^ =

j-n^2j-n^5j - 2ny2j+n c ^n+lOj - 5n^5j -2ny2j+n

jlOj-4nx Sj-2ny 2j +n £ R[Y>Y-1]

.

This completes the presentation of the example. Given a semigroup S— invariant if We

R [S ] - T[S]

investigate the class

distinguish a subclass is in ring

S , we say that

S*

implies S

S*

of all of

if each isomorphism

S

THEOREM 24.8. unitary ring Proof.

5*

The ring

R is

for any ring

T .

Z— invariant rings; we the ring

R

a:R[Z] -T[Z], for any 1 .

Part

is a subclass of R[Z]

is

(1) of Theorem 5 .

Z— invariant for each

R . Let

? <|>:R[Z ] -- h - T[Z]

be

Theorem 24.2, it suffices to show that assumption that



is elementary.


- T

as follows:

T , is elementary of degree

24.3 shows that, indeed,

R

the ring

, where

is nilpotent.

an isomorphism. R [Z ] = T

Thus, for u(g)

The mapping

easily seen to be a homomorphism of

Z

2

under the

g e Z

2

write

is a unit of

T

g — ► h(g)

is

into

By

Z .

If

and

346 h(g) = 0 <J>

for each

has degree

0

? R[Z ]

g , then viewing and Theorem 24.3

as

R[Z][Z]

implies that

,

R [Z ] = T .

2 Otherwise, Z = C g © Cg2^ » where Cg2) = ^er J1 • Considering 2 R[Z ] asR [ (g^) ] [ (g2) ] ,

R[Z2]

to

T[Z] , and again

T = R[(g^)]

~ R [Z ]

COROLLARY 24.9.

The following conditions are equivalent.

(1) R [Z ] - T [Z ]

.

(2) R[Zn ] = T[Zn ]

for each

n e

Z+ .

(3) R[Zn ] = T[Zn ]

for some

n e

Z+ .

Proof.

The implications

are clear.

To show that

(1) = >

(3) = >

and

Thus

R[Zn ] = R[Zn "2][Z2] = T[Zn ] = T[Zn _1 ] [Z]

n > 1 , implies

Assume that

Hence

ring, so R[Z] each

R

Z .

If it is the

and Theorem 24.1 shows that

K .Then

is reduced, and Z[G © Z]

is also

ufg)

G

G = H ©

is an isomor­


The mapping

and

is

11.6.

R .

H -Z

Z [G ]

by Theorem

unit of

Imfh) =

implies, by

is also a reduced indecomposable

has only trivial units

g e G , we write

into

Z[G © Z]

of Corollary 10.17 shows that

indecomposable.

.

G .

<J>:Z[G © Z] — -»-R[Z]

Theorem 9.17 shows that (1)

to show

R[Zn *] = T [ Z n *]

The integral group ring

Z— invariant for each abelian group

phism.

(3)

R[Zn '2][Z] = R[Zn_1 ] = T[Zn “1] .

COROLLARY 24.10.

Proof.

(2) = >

(1) , it suffices

R[Zn ] = T[Zn ]

Theorem 24.7, that

, for

(2)

that

part

.

H

Z [G ] -

is

For is a

g ----- ► h(g) zero mapping, then Z [G ] - R .

If

(Z[G]) £ R

h * 0 , then

a direct summand of Z[K][Z]

is

G -----

Z— invariant by

say

is

347 Theorem 24.8, and hence THEOREM 24.11. radical

J(R)

Z [G ] ^ R .

Let

R

be a unitary ring with Jacobson

and nilradical

N(R) .

Let

S

and S*

be as

in the paragraph preceding the statement of Theorem 24.8, (1)

If

R

is a

(2)

If

R

is an integral domain and if

then

R e S* (3)

R c S*

then

R e S* . J(R)* (0) ,

.

If

R is indecomposable and

minimal prime (4)

field,

P

of

R

such

if there exists

that R/P e S* ,

then

If

R is indecomposable and

if

If

R is quasi— local, then

R e S* .

a R e S*

J(R) * N(R)

.

, then

.

(5) (6)

If

dim R =

(7)

Each of the

0 , then

R e S* .

classes

S*

and S

isclosed under

taking finite direct sums. Proof.

We assume throughout the proof that

:R[Z] -- m - T[Z]

is an isomorphism.

composable in each of cases implies that the ring

T

The ring

R

is inde­

(1) — (5) , and Corollary 10.17

is also indecomposable.

Thus

<J>

is elementary in each of these cases --- say

<|>(X) =

uXr + t , and

= 1 . We begin

ourobject is to show that

|r|

to consider the individual cases. (1):

If

R

is a field,

tegral domains, and hence T[Z]

a - 1

R = GF(2)

(a) = vXm

.

If

We show that IRI > 2 , take

for some unit

is a unit of

and

T[Z]

is also a domain.

has only trivial units.

is clear if Then

T

then R[Z]

R , either

v e T

.

vXm + 1

Therefore


.

This

a e R \ {0} .

Since or

are in ­

a + 1

vXm - 1

or is a

348 unit of that

T[Z]

m =

, hence trivial.

0 , so <J>(a)e T

.

In either case we conclude

Thus

T[Z] =

<J>CR [Z ]) c T[uXr ,u lX r ] = T[Xr ,X r ] ,

and this implies

that

Ir | = 1 . (2):

As in (1) ,
show that unit of

R , so

some integer

T[Z]

£ T

.

has only trivial units.

Take

r e J (R)

1 + <|>(r) = uXm

m e Z .

m = 0 .

CyDCr)

and

T

and

<J>(R) c T

in

e T

with

R .

(r) e T\{0}

.

T[Z]

<j>(r) e T

.

The proof that

J (R)

Since <J>(y)

|r| = 1

,

and

Then yr e

,

c T[Z]

is an integral domain, it then follows that


is then completed as

(1) . (3):

<j>(P[Z])

Consider the minimal prime is minimal in

some minimal prime isomorphism since



(4): P

and

is also a unit of

y e

is a

=

Consequently,

(j)(J(R)) £ T . Take any element

1 + r u e T

2 <J>(l + r + r )

uXm + (uXm - l)2 = uX 2m - uXm + 1

so

Then

for some unit

Moreover,

and this implies that

.

We first

of

R

<|>*

is

of

of

, and hence

T

.

Since J (R)

onto

But

* N(R)

of

R[Z]

<J>

Then

since

for

induces an |

*|

(T/Q) [Z] and

|<J>*|= 1

.

<J>(P[Z]) = Q [Z ]

The isomorphism

(R/P)[Z]

elementary.

R/P

e S*

, there exists a minimal

.

prime

that is not expressible as an intersection of m a x ­

imal ideals of Jacobson

Q

T[Z]

P[Z]

R .

Hence

R/P

R/P e S *

radical, so

is a domain with nonzero by

(2) .

R e S*

Whence

by

(3) . (5):

Let

M

be the maximal ideal of

the result follows from minimal prime of (6):

R

(4) .

R .

In the contrary case,

and the result follows from

In this case


If

(1)

M * N(R) , M

is a

and

(3).

need not be elementary, but there

349 exist decompositions that

R = R_©...@R I n

= |R [z]

onto

T^[Z] S*

T = T,©...©T I n

an elementary isomorphism of

for each

and since

and

i .

Since

dim R^ = 0

if

R

M is a proper prime of

and

|<J>| = |<{>* |

R/M e S* , we (7):

have

We

for some since

Let

is in

for each

phism

e^

i

be

(Z?u.)X + 1 l zjt. e T[Z]

onto

S*

R.

l

T



Because

R^

belongs to R^ .

by Corollary 10.8, T^ =

induces

- T. for

an isomor­

If

R^ e

S

each i andthat

i

each R^is in

S* , then

is a unit of

T^

Thus

zl?u. is a unit of 1 l

nilpotent.

T ;

simultaneously.

T^[Z] for each i .

u^

of

the identity element of

nilpotent for each i .

z?t. where 1 i ’ is

S and

.It follows that

+ t^ , where

is

is

.

are in

On the other hand, if

^(e^X) = u^X

M

P

T = T^©. . .©Tn , where

for eachi , we conclude that *

t^ e T^[Z]

|<J>|

T[Z]

T and

of R^[Z]

R - T .

is elementary.

1 = |<J>* | =

Since the idempotents of

T(e^)

<J>

R = R^@...©Rn , where each

a specified class.

<^(ei)

is ele­

R , then

minimal prime

treat the cases of

Thus, assume that

each



induces an isomorphism

*: (R/M) [Z ] -- (T/P) [Z ] moreover,

i

is closed under taking finite direct sums by

Thus,

minimal in

R^[Z]

for each

(7) , we assume without loss of generality that mentary.

such


T

Consequently,

and

and

R e S*

if each

R^ e S* , and this completes the proof of Theorem 24.11. Section 24 Remarks Much of the material in this section stems from the paper [87] of Krempa.

K. Yoshida has also considered con­

sequences of isomorphism of

R[Z]

and

S[Z]

in [134]

.

In

350 particular, Yoshida also gives an example of non— isomorphic two-dimensional affine domains that

R[Z]

and

S[Z]

R

and

are isomorphic.

vestigates the class of

S

over a field such

Vleduc in [130] in­

S— invariant rings, where

S

is a

torsion— free cancellative monoid with quotient group of finite torsion— free rank.

351 §25.

Monoids in Isomorphic Monoid Rings --- a Survey

The dual of the question considered in Sections 23 and 24 is that of determining relations between monoids S2

if the monoid rings

unitary ring S2

R

isomorphic?

ral

R [S^ ]

and

are isomorphic.

R [S 2]

In particular, are

Zq

R[

]

[Zq ]

and

R[

are isomorphic only

if

and

are abelian groups such that - R[G][K]

, but

H

G © H and

>~] [Z^] k

and

are

m , but

m = k .The same type

of example is available for group rings:

R[G][H]

S^

The question has a negative answer in gene­

for example,

and

and

over a fixed

isomorphic for arbitrary positive integers Zq

S^

if

G , H , and

K

= G © K , then K need not be

isomorphic.

(Theorem 24.8 provides an indirect proof of the fact that no example of this type is possible for

G = Z .)

of a different type is obtained as follows. is a finite abelian group of order such that n—

n

is a unit

root of unity.

of

Using

K

Assume that

n

abelian group of order and

G

G

n

and that

K

and K

contains

a primitive

the fact that

K[G]

mensional and reduced, it follows easily that the direct sum of

An example

copies of n , then

K .

If H

is a field

is zero-di­ K[G] - Kn ,

is another

K[H] - Kn ^ K [G ] , but

H

need not be isomorphic.

While this isomorphism problem has hardly been considered in the literature for monoids that are not groups, the situa­ tion for group rings is quite different.

In fact, the

isomorphism question and the closely related problem of d e ­ termining the structure of the unit group of

R [G ]

were two

of the earliest topics concerning group rings considered. Because our treatment of units of

R[S]

in Section 11 was

352 abbreviated, many of the known results concerning the isomor­ phism problem are not accessible from results of this text. While some results are accessible,

this material is already

available in the books of Passman [121, Chapter 14] and Schgal

[125, Chapter 3], and it seemed pointless to include

proofs of inferior results here.

Hence,

this section merely

surveys some of the known results in the area, with a few references to the literature. We have already alluded to the paper

[76] of Higman in

which he determines, among other things, all finite abelian groups

G

such that the integral group ring

Z [G ]

has only

trivial units.

This result can be used to show that isomor­

phism of

and

Z[G]

Z[H]

, for finite abelian groups

H , implies isomorphism of R

and

H .

p— components of

G

GR

(resp.,

(resp.

H)

R , then isomorphism of

phism of

G/G r

and

HR )

0 , if

G

and

H

is the sum of the

such that

R[G]

and

More generally, if

is an integral domain of characteristic

are abelian, and if

in

G

G

and

p

R[H]

is invertible implies isomor­

H/HR ; the same conclusion holds for a

finitely generated indecomposable riijg of characteristic

0

(see [102-103]). If

F

is a unit of

is a field and

G

is a finite group whose order

F , then the group ring

rect sum of cyclotomic extensions of

F[G] F .

is a finite d i ­ Using an explicit

description of this decomposition in the case of the rational field

Q , it can be shown that

for finite abelian groups

G

field

F , isomorphism of

F[G]

if

is abelian and

G

F[G]

and

Q[G]

- Q [H ]

H . and

implies

G - H

For a more general

F[H]

implies

has only trivial units;

G = H if

F[G]

353 and

F[H]

are

F— isomorphic, then

isomorphic, where of

G

and

abelian then

T(G)

and

T(H)

H , respectively.

p— group and

F[G] = F[H]

F

G/T (G)

and

H/T(H)

are the torsion subgroups

Finally, if

G

is a countable

is a field of characteristic

implies

G - H .

are

p ,

Each of the results

mentioned in this paragraph can be found in [121] or in [125], In particular, much of the work on the isomorphism question for infinite abelian [18-19].

p— groups over fields is due to Berman

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II, Van Nostrand,

TOPIC INDEX

The following abbreviations are used in some of the listings in the index:

cong. for congruence, dom. for domain,

elt. for element, g p . for group, id. for ideal, mon.

for

monoid, and smgp. for semigroup. m a p . .75, 89, 102, 148, 279 Automorphism of a mon. ring.. 303, 306, 309

A Adjunction of an identity elt. canonical to a ring..104, 111 to a smgp...2, 275, 328, 331 Affine domain..289, 296, 299 Algebraic retract..147 Almost integral ring elt... 152 sm gp . e l t ...151 Annihilator of a ring elt... 94 Aperiodic smgp. elt.,.10, 326 Archimedean component of a smgp...236, 323, 326 decomposition of a smgp... 236, 322, 325 Ascending chain condition (a.c.c.) on cong...30, 39, 271, 272 on cyclic submonoids..181, 187, 191 on cyclic subgroups..181 on principal smgp. ideals.. 57 on principal ring ideals.. 171, 191 on smgp. ideals..40, 152 on subgroups..271 on submonoids..13 Associate elements of a mon... 52, 57 Augmentation ideal..75, 164, 277, 279, 338

B Basis of a free gp...l92, 309

219,

C Cancellation law for ring ideals..169 Cancellative semigroup elt...4 Canonical adjunction of identity elt. to a rin g..104, 111 form of a smgp. ring elt...

68 representation of a cong... 47 Center of a valuation on a g p ...221, 223 overring..194 Characteristic of a field..136, 254, 261, 279, 352 a ring. .98, 114, 118 , 125,' 136, 302 an integral dom...90 Chinese Remainder Theorem.. 243

363

364 Class g p . of a Dedekind dom. 343 Cohen's Structure Theorem.. 270 Comaximal ring ideals..164, 264 Complete integral closure.. 151, 153 Component homogeneous, of a graded r in g..85, 102 primary, of a group..100, 352 of a ring ideal..261, 266 Congruence(s) a.c.c. o n . .30, 271, 272 asymptotic equivalence..34, 97, 228, 320 cancellative..31, 44, 136, 227, 272, 324 canonical representation.. 47 commutative..318 d.c.c. o n . .271, 272 definition o f . .28 A . .35, 86 factor smgp. of..28 finitely generated..30, 72 generated by a set.. 29, 318 identity..29 irreducible..46 kernel id. of..79, 90, 99 lattice o f ..29, 71 on a g p ...33 p— equivalence..34, 97, 228, 320 primary..44 p r i me ..44 Ree s..36, 27 2 restricted to a subsemi­ group ..44 universal..29, 42 Content of a polynomial..293 smgp. ring elt...96, 113, 121, 131 Cosets of a subgroup..4 Cyclotomic field extension.. 352 D Dedekind-Mertens Lemma..96 Defining family for a Krull d om ...197 Degree of

a smgp. ring elt...84, 129 an elementary isomorphism.. 339 Descending chain condition (d.c.c.) on congruences..271, 272 on smgp. ideals..272 on subgroups..271 Dimension of a ring..226, 227, 288, 290, 328, 330, 343, 347 Direct sum of groups complete..25 external weak..26, 223 internal weak..77, 85, 100, 124, 183, 213, 272, 286, 321, 346, 351 into primary components.. 100 of m od ules..148, 149 of rings..100, 117, 123, 130, 142, 249, 250, 264, 266, 269, 308, 316, 338, 343, 347, 351 of semigroups external..18, 66, 180 weak internal..11, 61 Distributive law for ring ideals..243 Divisible hull of a group.. 163, 241 Divisor class group of a domain..209, 220, 223 of a m o n o i d ..219 Divisor class monoid of a d omain..208 of a m o n o i d ..217 Divisorial fractional ideal of a domain..208 of a m o n o i d ..215 E Element of a group divisible by an integer.. o f ^ y p e (0,0, ... ,0) . .180, 187 191 Element of a ring almost integral..152 annihilator of..94 idempotent..113, 308, 339 integral..152 nilpotent..117, 303, 305, 338

365 prime..183 u ni t..117, 303, 305 zero divisor..82, 89 Element of a smgp. or monoid almost integral..151 aperiodic..10, 326 associate..52,57 cancellative..4 divisor o f . .52 factor o f . .52 idempotent..11, 135, 231, 237 identity..2 index o f ..10, 229 integral..151 invertible..4, 52 irreducible..52, 56 multiple o f . .52 order o f ..10, 229 period o f ..10 periodic..10, 120, 326 p rime..52, 56 u ni t..52 Element of a smgp. ring or mon. ring canonical form of..68 content of..68, 96, 113, 121, 131 degree o f ..84, 129 G— component..134 I— component..134 idempotent..113, 135 invertible..120, 129, 131, 140 homogeneous..174 homogeneous content of..174 homogeneously primitive.. 174 mo ni c..84 monomial..89, 92, 113, 129, 172, 178 nilpotent..97 order o f ..84, 129 support of..68, 91, 113, 131, 138, 242 supporting smgp. of..68 zero divisor..82, 304 Element prime to a ring sub­ set . .17 3 Elementary isomorphism..338 degree o f ..339 Endomorphism injective..304, 306, 315 surjective..303, 306, 315 Exponent of a gp...267

F Factor smgp. of a congruence.. 28 Field algebraically closed..184 characteristic of..136, 254, 261, 279, 352 Fractional ideal of a domain divisorial..208 of a monoid..215 divisorial..215 principal..215 Free basis of a group..192, 219, 309 of a module..148, 150, 341 G Galois group..185 Gcd of smgp. elements..52, 54, 59, 175 Generating set for a smgp... 3 for a smgp. ideal..3 Gib of a family of congruences 29, 43, 72 Graded ring..85 homogeneous elt. of..85 Grading on a ring..85, 86, 174 Group a.c.c. on cyclic subgroups..181 cyclic submonoids..181 subgroups..271 cyclic..17, 168 d.c.c. on subgroups..271 divis ible..163 divisible hull of..163, 241 exponent o f ..267 finitely generated..41 free..192, 226 free basis of..192, 309 locally cyclic..16 mi x e d ..6 Noetherian..41 of divisor classes of a d o m . ..209, 220, 223 of a m o n ... 219 of invertible elts. of a mon. 120 p— quasicyclic..255

366 primary component of..280, 281 projection map on..192 quotient g p . of a cancella­ tive smgp...6, 93, 227, 237, 241, 288, 290 subgroups linearly ordered 256 torsion..6, 165 torsion— free..6, 165, 181 torsion— free rank of..165, 226, 280, 290, 350 torsion subgroup..121, 282 Group ring..88, 92, 120, 123, 127, 128 integral..346, 352 isomorphic..351 rational..352 H Height of a prime ideal..289, 293, 294 Homogeneous component of a graded ring..85, 102 an element..86 content of a mon. ring elt. 174 decomposition of an elt...86 elt. of a graded ri ng ..85 smg p. ring..174 Homogeneously primitive..174 Homomorphism of a smgp. ring 69 I Ideal of a ring contracted..149, 162, 243, 280, 330 finitely generated..72 idempotent..116, 121 invertible..243 maximal..330 minimal prime..212, 347, 348 multiplication..249 n i l . .116 primary..87, 212 p r i me ..83, 330 Ideal (s) of a smgp. or monoid 3, 36 d.c.c. on..272

prime..3, 40, 120, 136, 227, 273, 326 pr oper..3 radical of a n . .3, 43, 44 , 233, 323 Ideal of a smgp. ring augmentation..75, 164, 277, 279, 338 idempotent..320 kernel, of a cong...70, 90, 99, 271 Idempotent elt. of a ring..113, 308, 339 smgp...11, 135, 231, 237 smgp. ring..113, 135 Identity elt. of a mon...2 Imbedding a cancellative smgp. in a g p . 5, 7 smgp. in a monoid..2 Index of a smgp. elt...10, 229 Injective endomorphism..304, 306, 315 Integer regular on a ring..110 Integral closure of a mon...151 dependence in rings..166, 227, 291, 299 extension of a ring..125, 280 ring e l t ...152 smgp. element..151 Integral domain (or domain).. 82, 96 a.c.c.p. in..149, 171, 177, 188, 191 almost Dedekind..169, 189, 277, 285 a tomic..189 Bezout..162, 167, 240 characteristic of..90 complete integral closure.. 153 completely integrally closed 153, 209 Dedekind..162, 168, 240, 263 343 Euclidean..168 factorial..149, 171, 187, 196 GCD— domain..149, 171, 176 integral closure..158 integrally closed..147, 162 185, 206, 268 Krull domain..190, 198, 209 Noetherian..168, 206, 290 of finite character, etc. 206

367 II—domain . .207 PID..162, 168, 240, 267 Prufer..162, 167, 240, 302 valuation..169, 190, 209, 270 Integral group ring..346, 352 Irreducible monoid elt...52, 56 Isomorphism of group rings..351 monoid rings..337, 351 J Jacobian conjecture..315 Jacobson radical of a monoid ring..129, 133, 140 ri ng ..347 Jaffard’s Special Chain Theorem..288 K Kernel id. of a cong...70, 90, 99, 271 Krull domain..190, 198 defining family for..197 Krull Intersection Theorem 122, 128 L Lattice of congruences on a smgp...29, 38, 71 ideals of a ring..71 subgroups of a gp...7 5 Lem of ring elements..173 smgp. elements..53, 54, 59, 172 Lifting idempotents..116 Lub of a family of congruences 29, 35, 37, 72 Lying—over Theorem..280 Lying—under Theorem..186, 268 M McCoy's Theorem..84

Minimal prime of a ring ideal 186 Monic smgp. ring elt...84 Mono id ..2, 93, 119 a.c.c. on cyclic submonoids 181, 187, 191 aperiodic..135 cancellative..42, 52, 129, 148, 153, 162, 171, 212, 215, 218, 240, 242, 253, 289, 350 completely integrally closed 151, 214, 217 cyclic..167 divisor class mon...208, 217 factorial..57, 181, 187 finitely definable..76 finitely generated..168, 204, 289. finitely presented..76 free . .65, 77, 78 GCD-monoid..55, 171 integral closure..151, 156 integrally closed..151, 166, 206 Krull..190, 198 Noetherian..68, 76 numerical..12, 34 p e ri odi c ..260, 273 primitive..12 Prufer..166 quotient mon...6, 31 rank o f . .15 torsion— free..129, 153, 162, 171, 212, 218, 240, 289, 350 unique factorization..57 valuation..199 Monoid of divisors of a domain..208 of a mono id ..219 Monoid ring..67, 125, 129 arithmetical..167, 247, 253, 259, 261 Artinian..271, 274 as a graded ring..174 automorphisms of..303, 306, 309, 339 Bezout..167, 247, 253, 259, 263 chained..254, 256 completely integrally closed 155 definition of..67 dimension of..288 factorial..171, 187 GCD— domain..171, 176 general ZPI— ring..250, 253, 265

368

integrally closed..158, 159 isomorphism of..337, 351 elementary..338 degree o f ..339 Jacobson radical of..129, 133, 140 Krull domain..190 local..255 locally Noetherian..271, 280, 283 multiplication ring..250 nilradical o f . .97, 130, 140 Noetherian..75, 266, 271 P— ring..247 PIR..168, 250, 253, 265, 267 Priifer ring..167, 247 , 263 quasi— local..253, 255 reduced..138 RM— ri ng ..271, 277 satisfying a .c .c . p ...191, 271 semi—quasi— local..270 semilocal..270 trivial units of..312, 315, 347, 351 unit o f ..129, 131, 140, 167, 351 von Neumann regular..226, 228, 238, 259 Monomial..89, 92, 113, 129, 172, 178 p u r e ..205, 292 Multiplicative system in a ring..111 N Natural topology on a local ring..284 Nilpotent elt. of a ring..117, 303, 305, 338 smgp. rin g..97 Nilradical of a monoid ring..97, 130, 140 ring..89, 97, 131, 347 Noether Normalization Lemma.. 302 0 Order of nilpotency of a ring id.. 304, 311 a smgp. element..10, 229 a smgp. ring elt...84, 129 334

Order relation cardinal..26, 192, 203 lexicographic..26 partial..22 positive cone..24 reverse lexicographic..26 total..22, 153, 156 Orthogonal idempotents..141, 339 P Partial order definition o f ..22 on congruences..29 Period of a smgp. elt...10, 120 Periodic smgp. elt...10 Polynomial ring..64, 65, 78, 154, 159, 172, 226, 246, 288, 293, 303, 308, 317, 338, 351 Positive co n e ..24 Positive subset..24 Power series ring..21, 315, 336 Primary component of a group..100, 352 a ring ideal..261, 266 Primary Decomposition Theorem for congruences..48 for ring ideals..122 Prime element of a domain..183 a smg p... 52, 56 Prime subring..305 Projection map on a gp...l92 Pure difference binomial..78 monom ia l..205, 292 subgroup of a gp...l83

Q Quotient gp. of a cancellative smgp.. , 93, 227, 237, 241, 288, 290 monoid of a smgp...6, 31 overring of a domain..209 ring of a ring..Ill, 241, 243, 246, 253, 261

369 R

quasi— local..253 , 254 , 282 , 347 reduced..97, 132, 139, 156, 226, 312, 316, 320 R e e s ..344 restricted minimum condition 276, 286 S— invariant..345 second chain condition..286 semiprime..112 semisimple..129 special PIR (SPIR)..88, 249 , 264 special primary ring..249 von Neumann regular..113, 226, 228, 242, 259, 277, 328 Z— invariant..345 zero— dimensional..226, 271, 274 ZPI— ring..251 Root of unity..267, 332

Radical of a ring ideal..88 smgp. ideal..3, 43, 44, 233, 323 Rank of a m o n ... 15 Rank (torsion— free) of a gp... 165, 181, 226 Rational g p . ring..352 Relation asymmetric..22 compatible with smgp. opera­ tion ..22 partial order..22 total order..22 Ring arithmetical..240, 243, 251, 253, 257, 264 Artinian..257, 271, 286 Bezout..240, 244, 253 Boolean..113, 239 chain condition for primes.. 286 S chained, valuation..243, 253, 256, 270 Saturated chain condition..289 FMR-ring..103, 118, 139 general ZPI— ri ng ..240, 249, Semigroup..2 251, 253, 264 abelian..2 aperiodic..10, 119 graded by a smgp...85 Hilbert..102, 118, 139 Archimedean..233 cancellative..4, 7, 22, 31, idempotent..318 35, 44, 82, 106, 119, 129, idecomposable..113, 124, 126, 132, 312, 347, 352 236, 322 integrally closed..258 commutative..2 invariant..335 cyclic ..9, 114 local..256, 283 factor smgp. of a cong...28, 318 locally Cohen-Macaulay..286 finitely generated..68, 275 free of asymptotic torsion.. Gorenstein..286 Noetherian..277, 286 34, 105, 228, 236, 323 homomorphis.m of a.. 14 regular. .286 multiplication..240, 251 monogenic..9 Noetherian..30, 39 n— invariant..335 ni l. .98 numerical..34, 333 p— torsion— free..34, 99, 228, Noetherian..68, 75, 122,154, 249, 250, 255, 264, 302, 324 306, 309, 337, 343 partially ordered..22 periodic..7, 10, 227, 318, P—r i ng ..241, 247 PIR. .240, 253, 266 320, 328 polynomial..64, 65, 78, 154, separative..106 159, 172, 226, 246, 288, supporting..68 torsion— free..7, 22, 33, 35, 293, 303, 308, 317, 338, 82, 104, 129 351 totally ordered..22, 84 power invariant..336 Semigroup ring power series..21, 315, 336 Artinian..27 5 Prufer..240, 244, 251, 258

370 split by idempotents..141, 308 trivial..129, 133, 141, 144, 312, 315, 347, 352 ri ng ..117, 303, 305

definition o f ..64 homomorphism on..69 integral domain..82 isomorphism of..317, 331, 333 nilradical of..97, 320 Noetherian..77, 275 reduced..97, 320, 325 zero divisor..82, 304 Semilattice..231, 234, 239 Semilattice of subsemigroups.. 233 Shortest primary representation 264 Special chain of primes..288, 298 Split extension of subgroups.. 314 Subgroup maximal containing 0..4 of a smgp...2, 32, 93, 231 pure...183 pure subgroup generated by a s et ..183 torsion subgroup of a g p ... 121, 282 Subisomorphic rings..340, 342 Subsemigroup..2 Support of a smgp. ring elt... 68, 242 Surjective endomorphism..303, 306, 315 T

V v— operation on a domain..208 m on oi d..217 Valuation domain..169 center on a subring..194 discrete..190 essential..194 rank— one..190 Valuation monoid..200, 202,

220 essential..201 Valuation on a field..194, 200, 211 group..155, 220, 223 center..221, 223 discrete..190 essential..201 family of finite character 190 nontrivial..155 normed..203 rank o f ..190 trivial..155 Valuation overring..190 Value mon. of a g p . valuation 155, 190

Torsion— free rank of a g p ... 165, 181, 290, 350 Z Torsion subgroup of a gp.,.121, 282 Total quotient ring..149 Zero divisors of a Transcendence degree..289 ri n g ..82 Trivial units of a mon. ring.. s mg p. ring..82, 304 129, 133, 141, 144, 312, 315, 347, 352 Type of a g p . elt...180, 191, 212 U Unital extension of a ring..67, 68, 91, 93, 98, 111, 119 Units of a monoid ring..129, 131, 140, 169, 351 of finite order..143

INDEX OF MAIN NOTATION

SYMBOL S^

PAGE

MEANING

2

Monoid obtained element to S

by

adjoining an identity

A + B

2

Sum of subsets



3

Subsemigroup generated by

rad(I)

3

Radical of the ideal

Z

5

The integers

Zq

5

The set of nonnegative integers

c

7

Set inclusion

<

7

Proper inclusion of sets

A\B

7

Complement of

Q

7

Rational numbers

A,B of

B

S

I

in

B of

S

A

C(r,m)

10

Cyclic semigroup of index and period m

Z+

11

The positive integers

AS aeA a

11

Weak direct sum

11

Finite direct sum of

g.c.d.

12

Greatest common divisor

a.c.c.

13

Ascending chain condition

A @ B

18

External direct sum of

S/~

28

Factor semigroup with respect to congruence ~

29

Congruence congruence

S^.-.QSj^

£ ~2

371

of

is

r

{S } » a aeA S^,...,Sn

A

and

B the

less than or equal

to

372 SYMBOL

PAGE

MEANING

1. u .b .

29

Least upper bound

g.l.b.

29

Greatest lower bound

34

Congruence of

P A

35

p— equivalence

The minimal congruence with factor semigroup torsion— free and cancellative

P (s)

42

Congruence associated with

[p]°

42

An ideal of

[p]

42

The radical of

a |b

52

a

1 .c .m.

53

Least common multiple

a.c.c.p.

57

Ascending chain condition principal ideals

62

Cardinality of

R[X;S] or R [ S ]

64

The semigroup ring of

Supp(f)

68

The support of

c(f)

68

Content of

ker <J>*

69

The kernel of

72

least upper bound of congruences

72

greatest lower bound of congruences

77

Direct sum of

90

Characteristic of

Zn Char D PID

divides

S

p and

associated with

p

[p]°

b

for

A S

over

R

f

f *

n

copies of

Zq

D

165

Principal ideal domain

165

The torsion— free rank of

(g)

180

Cyclic subgroup generated

zw

185

Direct sum of

208

Divisorial ideal associated with

div (A)

208

The divisor class of

PCD)

208

The set of divisor classes

of

P(D)

209

Principal divisor classes

of

r0 (M)

Fv

s

w

M by

copies of

g Z F

A D D

373 SYMBOL

PAGE

MEANING

C (D)

209

Divisor class monoid of

I v

215

Divisorial ideal

of

div(I)

217

Divisor class of

I

PCS)

217

Divisors classes

of

PC S)

217

Principal divisor classes of

C(S)

217

Divisor class monoid of

SPIR

249

Special principal ideal ring

RS

317

Semigroup ring of noncommutative

2ir isi

317

Element of

deg
339

S

D

I e F(S)

S S

S

over

R,S

RS

Degree of an elementary isomorphism

INDEX OF SOME MAIN RESULTS

RESULT

PAGE

DESCRIPTION

1.1

3

Existence of prime ideals of a semigroup; description of rad(I)

2.1

9

Structure of cyclic semigroups

2.4

13

Numerical monoids are finitely generated

2.7

16

Q

2.9

17

Describes submonoids of

2.11

18

(R,#) finitely generated implies is finite

3.4

27

S can be ordered iff S is torsion— free and cancellative

4.1

28

Basic correspondence between congruences and homomorphisms

4.6

34

Equivalent conditions for asymptotic equivalence

4.8

35

The minimal congruence A such that S/A is torsion— free and cancellative

5.1

39

Basic properties of Noetherian semigroups

5.4

42

A cancellative Noetherian monoid is finitely generated

5.7

46

Each congruence on a Noetherain semigroup admits a primary decomposition

5.10

49

A Noetherian monoid is finitely generated

6.3

55

Equivalent conditions for a

6.5

57

Prime factorization in a monoid

is locally cyclic

375

Q R

GCD-monoid

376

RESULT

PAGE

DESCRIPTION

6.7

60

S is factorial iff S is a GCD-monoid satisfying a.c.c.p.

6.8

61

Structure of factorial monoids

7.2

69

Canonical induced homomorphisms on R[X;S] and their kernels

7.5

72

Fundamental properties of kernel ideals

7.7

75

R[X;S] is Noetherian iff R is Noetherian and S is finitely generated

7.8

76

A finitely generated monoid is Noetherian

7.9

76

Finitely generated implies finitely presented

7.11

78

Monoid rings expressed as homomorphic images of polynomial rings

8.1

82

Equivalent conditions for be an integral domain

R[X;S]

to

8.4

85

Zero divisors of a graded

8.6

87

Conditions under which divisor

f

is a zero

8.10

91

Subsets of S realizable support of a zero divisor

as the

8.13

94

1 - X s is a zero divisor periodic

9.1

97

How nilpotents of

9.9

104

N[ X ;S] if S

9.12

106

The nilradical of D[X;S] characteristic 0

9.13

106

S is separative iff asymptotic torsion

9.16

108

n{P.[X;S] + I R[X;S]

9.17 10.1

ring

iff

R [ X ;S ]

}

is

arise

is the nilradical is torsion— free

S

s

of

R [ X ;S]

, for

D

of

is free of

is the nilradical

of

px

110

Equivalent conditions for be reduced

113

How idempotents of

R [X ;S ]

R[X;S] arise

to

377

RESULT

PAGE

DESCRIPTION

10.6

118

Elements of Supp(f) , for idempotent, are periodic

10.9

120

Conditions under which idempotents of R[X;S] are in R

10.14

123

Equivalent conditions for each idempotent of R[ X;G ] to be in

f

R

10.17

126

R[X;S] is indecomposable iff R is indecomposable, the set G of periodic elements of S form a group,and the order of each nonzero element of G is a nonunit of R

11. 1

129

D [ X ;S ] has only trivial units if is torsion— free and cancellative

11.3

131

Equivalent conditions for to be a unit

11.6

133

Conditions under which only trivial units

11.13

140

If S is torsion— free and aperiodic, then f e R [X;S ] is a unit iff the G— component of f is a unit of R[X;G] and each coefficient of the (S\G)-compo­ nent of f is nilpotent

11.14

140

For S torsion— free andaperiodic, the Jacobson radical and nilradical of R[X;S] coincide

11.18

143

If H is a finitely generated group of units of R [X;S ] , then there exists a decomposition of 1 that splits each element of H

12.2

148

Relating the structure of distinguished subring R

12.4

152

The concepts of integrality and almost integrality in cancellative monoids

12.5

153

Complete integral closure of T [U]

12.7

155

D[S] iff

12.8

156

nGv

f e

S

R[X;S]

R[X;S]

has

T

to a

R [S ]

in

is completely integrally closed D and S are is the integral closure

of

S

a in

12.10

158

G

Integral closure of

R[S]

in

T[U]

378 RESULT

PAGE

DESCRIPTION

12.11

159

Integral closure of

13.6

167

Conditions for D[S] to be domain or a Bezout domain

13.8

168

D[S] is a Dedekind domain iff D a field and S is isomorphic to to

D[S] a Prufer

Z

is or

z0

14.1

171

D[S] a GCD-domain implies D is a GCD—domain and S is a GCD-monoid

14.5

176

The converse of Theorem 14.1

14.6

177

a.c.c.p.

14.8

178

D[S] are

14.16

187

Equivalent factorial

14.17

188

D[G] satisfies a.c.c.p. iff D satisfies a.c.c.p. and each nonzero element of G is of type (0,0,...)

15.1

191

Necessary conditions for a Krull domain

15.2

192

Characterization of Krull monoids

15.6

197

Equivalent conditions for a Krull domain

15.8

201

Uniqueness of a defining family of essential valuation monoids for a Krull monoid

15.9

202

Determination of the defining family for a Krull domain D[S]

16.2

211

C(D)

16.3

213

C(D) © C(K[S]) * C(D[S])

16.4

215

Basic properties of thev— operation on a monoid

16.7

219

C(S) * C(K[S])

16.10

223

A determination of Krull domain D[S]

17. 1

226

dim R[G] = dim I

in

D[S]

is factorial iff

D

and

conditions for

imbeds in

K[S]

D[S]

D[S]

D[S]

to be

to be

to be

C(D[S])

A | = r Q fG)

C(D[S])

for a

>

where

379

RESULT

PAGE

DESCRIPTION

17.3

227

dim R[S] = dim R

17.4

228

Equivalent conditions for (von Neumann) regular

17.9

234

Archimedean decomposition

17.10

236

The Archimedean components are cancel­ lative iff S is free of asymptotic torsion

18.5

242

If R[S] regular

18.6

244

R is arithmetical iff each chained

18.7

246

R regular implies ring

18.9

248

Equivalent conditions for R[S]to be a Priifer ring, a Bezout ring, or an arithmetical ring

18.10

250

The conditions PIR ,general ZPI— ring, and multiplication ring in R[S]

19.1

254

R[U] is quasi-local iff R is quasi— local, Char(R/M) = p * 0 U is a p— group

isa

iff

S

is periodic R[S] of

P— ring, then

R [X ]

to

be

S

R RM

is a

is is Bezout

and

19.4

256

Conditions

19. 7

259

Conditions for R [S ] to be arithmeti­ cal, where S is neither torsion— free nor periodic

19.13

265

R[S]

as a

ZPI— ring

19.16

267

D[G]

as a

PID

20. 5

273

If S satisfies a.c.c. and d.c.c. on congruences, then S is finite

20 . 6

274

R[S] and

S

for

R[U]

to be

is Artinian iff is finite

20.11

280

Necessary conditions for locally Noetherian

20.14

284

Converse of Theorem 20.11

21.4

290

dim R[S] = dim R[G] lative

if

chained

R is Artinian R[G]

S

is

to be

cancel­

380 RESULT

PAGE

DESCRIPTION

21.9

298

22.1

303

Determination of the of R[X]

22.3

306

R— automorphisms of

R[X,X *]

22.4

309

R— automorphisms of

R[Zn ]

23. 12

329

R [S* ] is the maximal regular subring of R[S]

23. 16

331

F[S] « K[T] periodic

23.17

333

F - K if aperiodic

24.2

338

Reduction to the case of elementary isomorphisms

24.6

342

R[Z] - T [ Z ] implies subisomorphic

24.9

346

R[ Z]

24.11

347

Sufficient conditions for isomorphism of R[Z] and T[Z] to imply that of R and T

Special Chain Theorem holds in R[S] for S torsion— free and finitely generated

implies

R— automorphisms

F

F [S] = K[T]

= T [ Z ] iff

R

R[Zn ]

= K and

if S S

and

is

is not

T are

= T[Zn ]