Rings and Categories of Modules / Edition 2

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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 familĀ­ 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, deĀ­ 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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Editorial Reviews

A textbook for introductory and more advanced graduate mathematics courses. Sets out most of the general theory of rings and modules for students who are familiar with rings from a standard undergraduate algebra course. The second edition (first in 1974) includes a new chapter containing many of the classical results on artinian rings that underlie the contemporary research. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9780387978451
  • Publisher: Springer New York
  • Publication date: 8/18/1998
  • Series: Graduate Texts in Mathematics Series, #13
  • Edition description: 2nd ed. 1992
  • Edition number: 2
  • Pages: 378
  • Product dimensions: 6.20 (w) x 9.40 (h) x 1.10 (d)

Table of Contents

0 Preliminaries 1
Ch. 1 Rings, Modules and Homomorphisms 10
1 Review of Rings and their Homomorphisms 10
2 Modules and Submodules 26
3 Homomorphisms of Modules 42
4 Categories of Modules; Endomorphism Rings 55
Ch. 2 Direct Sums and Products 65
5 Direct Summands 65
6 Direct Sums and Products of Modules 78
7 Decomposition of Rings 95
8 Generating and Cogenerating 105
Ch. 3 Finiteness Conditions for Modules 115
9 Semisimple Modules - The Socle and the Radical 115
10 Finitely Generated and Finitely Cogenerated Modules - Chain Conditions 123
11 Modules with Composition Series 133
12 Indecomposable Decompositions of Modules 140
Ch. 4 Classical Ring-Structure Theorems 150
13 Semisimple Rings 150
14 The Density Theorem 157
15 The Radical of a Ring - Local Rings and Artinian Rings 165
Ch. 5 Functors Between Module Categories 177
16 The Hom Functors and Exactness - Projectivity and Injectivity 178
17 Projective Modules and Generators 191
18 Injective Modules and Cogenerators 204
19 The Tensor Functors and Flat Modules 218
20 Natural Transformations 234
Ch. 6 Equivalence and Duality for Module Categories 250
21 Equivalent Rings 250
22 The Morita Characterizations of Equivalence 262
23 Dualities 269
24 Morita Dualities 278
Ch. 7 Injective Modules, Projective Modules, and Their Decompositions 288
25 Injective Modules and Noetherian Rings - The Faith-Walker Theorems 288
26 Direct Sums of Countably Generated Modules - With Local Endomorphism Rings 295
27 Semiperfect Rings 301
28 Perfect Rings 312
29 Modules with Perfect Endomorphism Rings 322
Ch. 8 Classical Artinian Rings 327
30 Artinian Rings with Duality 327
31 Injective Projective Modules 336
32 Serial Rings 345
Bibliography 363
Index 369
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