# multiplicative theory of ideals

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Multiplicative Theory of Ideals

This is Volume 43 i n PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: PAUL SMITH A. AND SAMUEL EILENBERGA complete list of titles in this series appears at the end of this volume

MULTIPLICATIVE THEORY OF IDEALSM A X D. LARSEN / PAUL J . McCARTHYUniversity of Nebraska Lincoln, Nebraska University of Kansas Lawrence, Kansas

A C A D E M I C P R E S S New York and London

@

1971

COPYRIGHT 0 1971, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS, INC.

111 Fifth Avenue, New York, New York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD.Berkeley Square House, London W l X 6BA

LIBRARY OF CONGRESS CATALOG CARD

NUMBER: 72-137621

AMS (MOS)1970 Subject Classification 13F05; 13A05,13B20, 13C15,13E05,13F20PRINTED IN THE UNITED STATES OF AMERICA

To Lillie and Jean

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ContentsPreface PrerequisitesxiXlll

...

Chapter I. Modules1 2 3 4 5Rings and Modules Chain Conditions Direct Sums Tensor Products Flat Modules Exercises

1 8 12 15 21 27

Chapter II.1 2 3 4

Primary Decompositions and Noetherian Rings36 39

Operations on Ideals and Submodules Primary Submodules Noetherian Rings Uniqueness Results for Primary Decompositions Exercises

44 48 52

Chapter Ill. Rings and Modules of Quotients1 Definition 2 Extension and Contraction of Ideals 3 Properties of Rings of Quotients ExercisesVii

61 66 71 74

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CONTENTS

Chapter IV.1 2 3 4

Integral Dependence82 84 88 92 94

Definition of Integral Dependence Integral Dependence and Prime Ideals Integral Dependence and Flat Modules Almost Integral Dependence Exercises

Chapter V.

Valuation Rings99 105 107 114 118

1 T h e Definition of a Valuation Ring 2 Ideal Theory in Valuation Rings 3 Vaiuations 4 Prolongation of ValuationsExercises

Chapter VI.

Priifer and Dedekind Domains124 126 132 134 140 144

1 Fractional Ideals 2 Prufer Domains 3 Overrings of Priifer Domains4

Dedekind Domains Exercises

5 Extension of Dedekind Domains

Chapter VII.

Dimension of Commutative Rings156 161 164 168

1 T h e Krull Dimension 2 T h e Krull Dimension of a Polynomial Ring 3 Valuative Dimension Exercises

Chapter VIII.

Krull Domains171 179 185 190 194

1 Krull Domains 2 Essential Valuations 3 The Divisor Class Group

4 Factorial RingsExercises

CONTENTS

ix

Chapter IX. Generalizations of Dedekind Domains1 2 3 4Almost Dedekind Domains ZPI-Rings Multiplication Rings Almost Multiplication Rings Exercises

201 205 209 216 220

Chapter X.

Prufer Rings226 232 234 236 244

1 Valuation Pairs2 Counterexamples 3 Large Quotient Rings 4 Prufer Rings Exercises

Appendix: Decomposition of Ideals in Noncommutative RingsExercises

252 263 266 294

BibliographySubject Index

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Preface

The viability of the theory of commutative rings is evident from the many papers on the subject which are published each month. This is not surprising, considering the many problems in algebra and geometry, and indeed in almost every branch of mathematics, which lead naturally to the study of various aspects of commutative rings. In this book we have tried to provide the reader with an introduction to the basic ideas, results, and techniques of one part of the theory of commutative rings, namely, multiplicative ideal theory. The text may be divided roughly into three parts. I n the first part, the basic notions and technical tools are introduced and developed. In the second part, the two great classes of rings, the Prufer domains and the Krull rings, are studied in some detail. In the final part, a number of generalizations are considered. In the appendix a brief introduction is given to the tertiary decomposition of ideals of noncommutative rings. The lengthy bibliography begins with a list of books, some on commutative rings and others on related subjects. Then follows a list of papers, all more or less concerned with the subject matter of the text. This book has been written for those who have completed a course in abstract algebra at the graduate level. Preceding the text there is a discussion of some of the prerequisites which we consider necessary. At the end of each chapter are a number of exercises. They are of three types. Some require the completion of certain technical details-they might possibly be regarded as busy work. Others xi

xii

PREFACE

contain examples-some of these are messy, but it will be beneficial for the reader to have some experience with examples. Finally, there are exercises which enlarge upon some topic of the text or which contain generalizations of results in the text-the bulk of the exercises are of this type. A number of exercises are referred to in proofs, and those proofs cannot be considered to be complete until the relevant exercises have been done. We wish to thank those of our colleagues and students who have commented on our efforts over the years. Special thanks goes to Thomas Shores for his careful reading of the entire manuscript, and to our wives for their patience.

Prerequisites

A graduate level course in abstract algebra will provide most of the background knowledge necessary to read this book. In several places we have used a little more field theory than might be given in such a course. The necessary field theory may be found in the first two chapters of Algebraic Extensions of Fields by McCarthy, which is listed in the bibliography. One thing that is certainly required is familiarity with Zorns lemma. Let S be a set. A partial ordering on S is a relation < on S such that

(i) s l s f o r a l l s E S ; (ii) if s < t and t I s , then s = t ; and (iii) if s < t and t < u, then s < u.The set S, together with a partial ordering on S, is called a partially ordered set. Let S be a partially ordered set. A subset T of S is called totally ordered if for all elements s, t E T either s < t or t < s. Let S be a subset of S. An element s E S is called an upperbound of S if s I all s E S. An element s E S is called a maximal element s for of S if for an element t ES, s 5 t implies that t = s. Note that S may have more than one maximal element.Zorns Lemma. Let S be a nonempty partially ordered set. If e v u y totally ordered subset of S has an upper bound in S, then S has a maximal element.Xlll

...

xiv

PREREQUISITES

If A and B are subsets of some set, then A s B means that A is a subset of B, and A c B means that A E B but A # B. If S is a set of subsets of some set, then S is a partially ordered set with 2 as the partial ordering. Whenever we refer to a set of subsets as a partially ordered set we mean with this partial ordering. Let S and T be sets and consider a mapping f : S+ T. The mapping can be described explicitly in terms of elements by writing sHf(s). If A is a subset of S, we write f ( A )= {f(s) \SEA},and if B is a subset of T , we write f - l ( B )= {s I S E S f(s) E B } .Thus, and f provides us with two mappings, one from the set of subsets of S into the set of subsets of T and another in the opposite direction. We assume that the reader can manipulate with these mappings. If S, T , and U are sets and f : S - t T and g : T+ U are mappings, their composition gf : S+ U is defined by (gf) (s) = g ( f ( s ) ) for all s E S. On several occasions we shall use the Kronecker delta a,,, which is defined by

1

if

i=j,

Multiplicative Theory of Ideals

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CHAPTER

Modules

1 RINGS A N D MODULES

We begin by recalling the definition of ring. A ring R is a nonempty set, which we aiso denote by R, together with two binary operations (a, b) H U b and (a, b) H ab (addition and multiplication, respectively), subject to the following conditions :

+

(i) the set R, together with addition, is an Abelian group; (ii) a(bc) = (ab)c for all a, b, c E R ; (iii) a(b + c) = ab ac and (b c)u = ba ca for all a, b, c E R.

+

+

+

Let R be a ring, T h e identity element of the group of (i) will be denoted by 0 ; the inverse of an element a E R considered as an element of this group will be denoted by - a ; a+(-b) will be written a - b. T h e reader may verify for himself such statements asOa = a0 = 0 a( -b) = (-a)b = -(ab)

a(b -c)

= ab -uc

for all a E R, for all a, b E R, for all a, b, c E R.

A ring R is said to be commutative if ab = ba for all a, b E R. An element of R is called a unity, and is denoted by 1, if la = a1 = a for all a E R. If R has a unity, then it has exactly one unity. We shall assume throughout this book that all rings under consideration have unities. By a subring of a ring R we mean a ring S such that the1

2

1

MODULES

set S is a subset of the set R and such that the binary operations of R yield the binary operations of S when restricted to S x S . By our assumption concerning unities, both R and S have unities. We shall consider only those subrings of a ring R which have the same unity as R. By a left ideal of a ring R we mean a nonempty subset A of R such that a - b E A and ra E A for all a, b E A and r E R. By a right ideal of R we mean a nonempty subset B of R such that a - b E B and ar E B for all a, b E B and r E R. A left ideal of R which is at the same time

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