附註:Includes bibliographical references and indexes.
Cover -- Table of Contents -- Preface -- Chapter 1. Preliminaries -- 1.1 Basic concepts and examples -- 1.2 Modules and homomorphisms -- 1.3 Classical isomorphism theorems -- 1.4 Direct sums and products -- 1.5 Finitely generated and free modules -- 1.6 Notes and references -- Chapter 2. Decompositions of rings -- 2.1 Two-sided Peirce decompositions of a ring -- 2.2 The Wedderburn-Artin theorem -- 2.3 Lattices. Boolean algebras and rings -- 2.4 Finitely decomposable rings -- 2.5 Notes and references -- Chapter 3. Artinian and Noetherian rings -- 3.1 Artinian and Noetherian modules and ring -- 3.2 The Jordan-Hèolder theorem -- 3.3 The Hilbert basis theorem -- 3.4 The radical of a module and a ring -- 3.5 The radical of Artinian rings -- 3.6 A criterion for a ring to be Artinian or Noetherian -- 3.7 Semiprimary rings -- 3.8 Notes and references -- Chapter 4. Categories and functors -- 4.1 Categories, diagrams and functors -- 4.2 Exact sequences. Direct sums and direct products -- 4.3 The Hom functors -- 4.4 Bimodules -- 4.5 Tensor products of modules -- 4.6 Tensor product functor. -- 4.7 Direct and inverse limits -- 4.8 Notes and references -- Chapter 5. Projectives, injectives and flats -- 5.1 Projective modules -- 5.2 Injective modules -- 5.3 Essential extensions and injective hulls -- 5.4 Flat modules -- 5.5 Right hereditary and right semihereditary rings -- 5.6 Herstein-Small rings -- 5.7 Notes and references -- Chapter 6. Homological dimensions -- 6.1 Complexes and homology. Free resolutions -- 6.2 Projective and injective resolutions. Derived functors -- 6.3 The functor Tor -- 6.4 The functor Ext -- 6.5 Projective and injective dimensions -- 6.6 Global dimensions -- 6.7 Notes and references -- Chapter 7. Integral domains -- 7.1 Principal ideal domains -- 7.2 Factorial rings -- 7.3 Euclidean domains. -- 7.4 Rings of fractions and quotient fields -- 7.5 Polynomial rings over factorial rings -- 7.6 The Chinese remainder theorem -- 7.7 Smith normal form over a
摘要:The text of the first volume of the book covers the major topics in ring and module theory and includes both fundamental classical results and more recent developments. The basic tools of investigation are methods from the theory of modules, which allow a very simple and clear approach both to classical and new results. An unusual main feature of this book is the use of the technique of quivers for studying the structure of rings. A considerable part of the first volume of the book is devoted to a study of special classes of rings and algebras, such as serial rings, hereditary rings, semidistr.
