# Torsion-Free Modules

University of Chicago Press, 1972 - Mathematics - 168 pages
The subject of torsion-free modules over an arbitrary integral domain arises naturally as a generalization of torsion-free abelian groups. In this volume, Eben Matlis brings together his research on torsion-free modules that has appeared in a number of mathematical journals. Professor Matlis has reworked many of the proofs so that only an elementary knowledge of homological algebra and commutative ring theory is necessary for an understanding of the theory.

The first eight chapters of the book are a general introduction to the theory of torsion-free modules. This part of the book is suitable for a self-contained basic course on the subject. More specialized problems of finding all integrally closed D-rings are examined in the last seven chapters, where material covered in the first eight chapters is applied.

An integral domain is said to be a D-ring if every torsion-free module of finite rank decomposes into a direct sum of modules of rank 1. After much investigation, Professor Matlis found that an integrally closed domain is a D-ring if, and only if, it is the intersection of at most two maximal valuation rings.

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### Contents

 Introduction 1 Cotorsion modules 9 Completions 17 hlocal rings 26 Reflexive rings 37 Noetherian reflexive rings 47 Torsionless rings 57 Completely reflexive rings 64
 Noetherian Drings 97 Quasilocal Drings 102 hlocal Drings 115 Rings of type I 125 Integrally closed Drings 141 Hausdorff Drings 156 Conclusion 162 Bibliography 164

 Maximal valuation rings 73 The two generator problem for ideals 84

### Popular passages

Page 164 - IN = 0. From this it follows immediately that F is a complete discrete valuation ring. Let x be an element of F such that N = Fx. Every ideal of F is a power of N. Now dim , F/N < dim /F/FM < 2.

### References to this book

 Modules over Valuation RingsLimited preview - 1985
 Non-Noetherian Commutative Ring TheoryLimited preview - 2000
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