Rings and Categories of Modules

Description

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. 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 Back Matter....Pages 327-339