Rings and Categories of Modules

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Springer Science & Business Media, Dec 6, 2012 - Mathematics - 378 pages
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This book is intended to provide a reasonably self-contained account of a major portion of the general theory of rings and modules suitable as a text for introductory and more advanced graduate courses. We assume the famil iarity with rings usually acquired in standard undergraduate algebra courses. Our general approach is categorical rather than arithmetical. The continuing theme of the text is the study of the relationship between the one-sided ideal structure that a ring may possess and the behavior of its categories of modules. Following a brief outline of set-theoretic and categorical foundations, the text begins with the basic definitions and properties of rings, modules and homomorphisms and ranges through comprehensive treatments of direct sums, finiteness conditions, the Wedderburn-Artin Theorem, the Jacobson radical, the hom and tensor functions, Morita equivalence and duality, de composition theory of injective and projective modules, and semi perfect and perfect rings. In this second edition we have included a chapter containing many of the classical results on artinian rings that have hdped to form the foundation for much of the contemporary research on the representation theory of artinian rings and finite dimensional algebras. Both to illustrate the text and to extend it we have included a substantial number of exercises covering a wide spectrum of difficulty. There are, of course" many important areas of ring and module theory that the text does not touch upon.
 

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Contents

Preface
1
Rings Modules and Homomorphisms
10
2 Modules and Submodules
26
3 Homomorphisms of Modules
42
4 Categories of Modules Endomorphism Rings
55
Direct Sums and Products
65
5 Direct Summands
77
6 Direct Sums and Products of Modules
93
12 Indecomposable Decompositions of Modules
140
Classical RingStructure Theorems
150
14 The Density Theorem
157
15 The Radical of a RingLocal Rings and Artinian Rings
165
Functors Between Module Categories
177
16 The Hom Functors and ExactnessProjectivity and Injectivity
178
17 Projective Modules and Generators
203
Equivalence and Duality for Module Categories
250

7 Decomposition of Rings
103
8 Generating and Cogenerating
113
Finiteness Conditions for Modules
115
9 Semisimple ModulesThe Socle and the Radical
121
10 Finitely Generated and Finitely Cogenerated Modules Chain Conditions
123
11 Modules with Composition Series
133
Injective Modules Projective Modules
288
Classical Artinian Rings
327
Bibliography
363
Index
369
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