Ebook: Rings and Categories of Modules

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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This book is intended to provide a self-contained account of much of the theory of rings and modules. The theme of the text throughout is the relationship between the one-sided ideal structure a ring may possess and the behavior of its categories of modules. Following a brief outline of the foundations, the book begins with the basic definitions and properties of rings, modules and homomorphisms. The remainder of the text gives comprehensive treatments of direct sums, finiteness conditions, the Wedderburn-Artin Theorem, the Jacobson radical, the hom and tensor functions, Morita equivalence and duality, decomposition theory, and semiperfect and perfect rings. This second edition includes a chapter containing many of the classical results on Artinian rings that have helped form the foundation for much of contemporary research on the representation theory of Artinian rings and finite-dimensional algebras.

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This book is intended to provide a self-contained account of much of the theory of rings and modules. The theme of the text throughout is the relationship between the one-sided ideal structure a ring may possess and the behavior of its categories of modules. Following a brief outline of the foundations, the book begins with the basic definitions and properties of rings, modules and homomorphisms. The remainder of the text gives comprehensive treatments of direct sums, finiteness conditions, the Wedderburn-Artin Theorem, the Jacobson radical, the hom and tensor functions, Morita equivalence and duality, decomposition theory, and semiperfect and perfect rings. This second edition includes a chapter containing many of the classical results on Artinian rings that have helped form the foundation for much of contemporary research on the representation theory of Artinian rings and finite-dimensional algebras.
Content:
Front Matter....Pages i-ix
Preliminaries....Pages 1-9
Rings, Modules and Homomorphisms....Pages 10-64
Direct Sums and Products....Pages 65-114
Finiteness Conditions for Modules....Pages 115-149
Classical Ring-Structure Theorems....Pages 150-176
Functors Between Module Categories....Pages 177-249
Equivalence and Duality for Module Categories....Pages 250-287
Injective Modules, Projective Modules, and Their Decompositions....Pages 288-326
Classical Artinian Rings....Pages 327-361
Back Matter....Pages 363-378

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This book is intended to provide a self-contained account of much of the theory of rings and modules. The theme of the text throughout is the relationship between the one-sided ideal structure a ring may possess and the behavior of its categories of modules. Following a brief outline of the foundations, the book begins with the basic definitions and properties of rings, modules and homomorphisms. The remainder of the text gives comprehensive treatments of direct sums, finiteness conditions, the Wedderburn-Artin Theorem, the Jacobson radical, the hom and tensor functions, Morita equivalence and duality, decomposition theory, and semiperfect and perfect rings. This second edition includes a chapter containing many of the classical results on Artinian rings that have helped form the foundation for much of contemporary research on the representation theory of Artinian rings and finite-dimensional algebras.
Content:
Front Matter....Pages i-ix
Preliminaries....Pages 1-9
Rings, Modules and Homomorphisms....Pages 10-64
Direct Sums and Products....Pages 65-114
Finiteness Conditions for Modules....Pages 115-149
Classical Ring-Structure Theorems....Pages 150-176
Functors Between Module Categories....Pages 177-249
Equivalence and Duality for Module Categories....Pages 250-287
Injective Modules, Projective Modules, and Their Decompositions....Pages 288-326
Classical Artinian Rings....Pages 327-361
Back Matter....Pages 363-378
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