Torsion-Free Modules

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University of Chicago Press, 1972 - Mathematics - 168 pages
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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

Common terms and phrases

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.

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