Multiplicative Theory of Ideals This is Volume 43 in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: PAULA. SMITHAND SAMUELEILENBERG A complete list of titles in this series appears at the end of this volume MULTIPLICATIVE THEORY OF IDEALS MAX D. LARSEN / PAUL J. McCARTHY University of Nebraska University of Kansas Lincoln, Nebraska Lawrence, Kansas @ A CADEM I C P RE S S New York and London 1971 COPYRIGHT 0 1971, BY ACADEMICPRESS, 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 WlX 6BA LIBRARY OF CONGRESS CATALOG CARD NUMBER: 72-137621 AMS (MOS)1970 Subject Classification 13F05; 13A05,13B20, 13C15,13E05,13F20 PRINTED IN THE UNITED STATES OF AMERICA To Lillie and Jean This Page Intentionally Left Blank Contents Preface xi ... Prerequisites Xlll Chapter I. Modules 1 Rings and Modules 1 2 Chain Conditions 8 3 Direct Sums 12 4 Tensor Products 15 5 Flat Modules 21 Exercises 27 Chapter II. Primary Decompositions and Noetherian Rings 1 Operations on Ideals and Submodules 36 2 Primary Submodules 39 3 Noetherian Rings 44 4 Uniqueness Results for Primary Decompositions 48 Exercises 52 Chapter Ill. Rings and Modules of Quotients 1 Definition 61 2 Extension and Contraction of Ideals 66 3 Properties of Rings of Quotients 71 Exercises 74 Vii Vlll CONTENTS Chapter IV. Integral Dependence 1 Definition of Integral Dependence 82 2 Integral Dependence and Prime Ideals 84 3 Integral Dependence and Flat Modules 88 4 Almost Integral Dependence 92 Exercises 94 Chapter V. Valuation Rings 1 The Definition of a Valuation Ring 99 2 Ideal Theory in Valuation Rings 105 3 Vaiuations 107 4 Prolongation of Valuations 114 Exercises 118 Chapter VI. Priifer and Dedekind Domains 1 Fractional Ideals 124 2 Prufer Domains 126 3 Overrings of Priifer Domains 132 4 Dedekind Domains 134 5 Extension of Dedekind Domains 140 Exercises 144 Chapter VII. Dimension of Commutative Rings 1 The Krull Dimension 156 2 The Krull Dimension of a Polynomial Ring 161 3 Valuative Dimension 164 Exercises 168 Chapter VIII. Krull Domains 1 Krull Domains 171 2 Essential Valuations 179 3 The Divisor Class Group 185 4 Factorial Rings 190 Exercises 194 CONTENTS ix Chapter IX. Generalizations of Dedekind Domains 1 Almost Dedekind Domains 201 2 ZPI-Rings 205 3 Multiplication Rings 209 4 Almost Multiplication Rings 216 Exercises 220 Chapter X. Prufer Rings 1 Valuation Pairs 226 2 Counterexamples 232 3 Large Quotient Rings 234 4 Prufer Rings 236 Exercises 244 Appendix: Decomposition of Ideals in Noncommutative Rings 252 Exercises 263 Bibliography 266 Subject Index 294 This Page Intentionally Left Blank 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. In 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 Zorn’s lemma. Let S be a set. A partial ordering on S is a relation < on S such that (i) slsforallsES; (ii) if s<t and t Is, 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 upper‘bound of S’ if s‘ Is for all s’ E S’. An element s E S is called a maximal element 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. Zorn’s Lemma. Let S be a nonempty partially ordered set. If evuy 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 ISESand f(s) EB}.Thus, 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 map- pings, 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 This Page Intentionally Left Blank CHAPTER Modules 1 RINGS AND MODULES We begin by recalling the definition of ring. A ring R is a non- empty set, which we aiso denote by R, together with two binary operations (a,b) HU + 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, The 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. The reader may verify for himself such statements as Oa = a0 = 0 for all a E R, a( -b) = (-a)b = -(ab) for all a, b E R, a(b -c) = ab -uc 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 the 1 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.