## Rings and Categories of ModulesThis 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

1 | |

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 |

363 | |

369 | |

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### Common terms and phrases

abelian group algebra annihilator artinian ring BiBnd(RM bimodule commutative complements direct summands complements maximal composition series Corollary decomposition that complements denote diagram direct sum direct summand division ring dual element End(RM endomorphism ring epic epimorphism exact sequence Exercise exists factor module faithful finitely cogenerated finitely generated projective following are equivalent functors Hint Home Homs Homs(N idempotents indecomposable decomposition indexed set injective envelope injective modules inverse Ker f lattice left ideal left noetherian Lemma let f matrices maximal submodule minimal module RM monic monomorphism Morita duality morphism multiplication natural isomorphism nilpotent orthogonal primitive idempotents primitive ring projective cover projective module Proof Proposition Prove quasi-regular R-homomorphism right artinian right R-module ring and let ring homomorphism semiperfect ring semisimple modules serial ring simple left R-module simple modules subring subset Suppose Theorem U-reflexive unique vector space