Groups, Rings, Modules

Description

The main thrust of this book is easily described. It is to introduce the reader whoalready has some familiarity with the basic notions of sets, groups, rings, andvector spaces to the study of rings by means of their module theory. This programis carried out in a systematic way for the classicalJy important semisimple rings,principal ideal domains, and Oedekind domains. The proofs of the well-knownbasic properties of these traditionally important rings have been designed toemphasize general concepts and techniques. HopefulJy this wilJ give the reader agood introduction to the unifying methods currently being developed in ringtheory.CONTENTS Preface ix PART ONE 1 Chapter I SETS AND MAPS 3 I. Sets and Subsets 32. Maps S 3. Isomorphisms of Sets 7 4. Epimorphisms and Monomorphisms 8 S. The Image Analysis of a Map 106. The Coimage Analysis of a Map II7. Description of Surjective Maps 128. Equivalence Relations 13 9. Cardinality of Sets IS 10. Ordered Sets 16II. Axiom of Choice 17 12. Products and Sums of Sets 20Exercises 23Chapter 2 MONOIDS AND GROUPS 271. Monoids 272. Morphisms of Monoids 303. Special Types of Morphisms 324. Analyses of Morphisms 375. Description of Surjective Morphisms 396. Groups and Morphisms of Groups 417. Kernels of Morphisms of Groups 438. Groups of Fractions 499. The Integers 5510. Finite and Infinite Sets 57Exercises 64Chapter 3 CATEGORIES 751. Categories 752. Morphisms 793. Products and Sums 82Exercises 85Chapter 4 RINGS 991. Category of Rings 992. Polynomial Rings 1033. Analyses of Ring Morphisms 1074. Ideals 1125. Products of Rings 115Exercises 116PART TWO 127Chapter 5 UNIQUE FACTORIZATION DOMAINS 129I. Divisibility 1302. Integral Domains 1333. Unique Factorization Domains 1384. Divisibility in UFD's 1405. Principal Ideal Domains 1476. Factor Rings of PID's 1527. Divisors 1558. Localization in Integral Domains 1599. A Criterion for Unique Factorization 16410. When R [X] is a UFD 169Exercises 171Chapter 6 GENERAL MODULE THEORY 1761. Category of Modules over a Ring 1782. The Composition Maps in Mod(R) 1833. Analyses of R-Module Morphisms 1854. Exact Sequences 1935. Isomorphism Theorems 2016. Noetherian and Artinian Modules 2067. Free R-Modules 2108. Characterization of Division Rings 2169. Rank of Free Modules 22110. Complementary Submodules of a Module 22411. Sums of Modules 231CONTENTS vII12. Change of Rings 23913. Torsion Modules over PID's 24214. Products of Modules 246Exercises 248Chapter 7 SEMISIMPLE RINGS AND MODULES 266I. Simple Rings 2662. Semisimple Modules 2713. Projective Modules 2764. The Opposite Ring 280Exercises 283Chapter 8 ARTINIAN RINGS 2891. Idempotents in Left Artinian Rings 2892. The Radical of a Left Artinian Ring 2943. The Radical of an Arbitrary Ring 298Exercises 302PART THREE 311Chapter 9 LOCALIZATION AND TENSOR PRODUCTS 3131. Localization of Rings 3132. Localization of Modules 3163. Applications of Localization 3204. Tensor Products 3235. Morphisms of Tensor Products 3286. Locally Free Modules 334Exercises 337Chapter 10 PRINCIPAL IDEAL DOMAINS 351I. Submodules of Free Modules 3522. Free Submodules of Free Modules 3553. Finitely Generated Modules over PID's 3594. Injective Modules 3635. The Fundamental Theorem for PID's 366Exercises 371Chapter II APPLICATIONS OF FUNDAMENTAL THEOREM 376I. Diagonalization 3762. Determinants 3803. Mat rices 3874. Further Applications of the Fundamental Theorem 3915. Canonical Forms 395Exercises 40 IPART FOUR 413Chapter 12 ALGEBRAIC FIELD EXTENSIONS 4151. Roots of Polynomials 4152. Algebraic Elements 4203. Morphisms of Fields 4254. Separability 4305. Galois Extensions 434Exercises 440Chapter 13 DEDEKIND DOMAINS 445I. Dedekind Domains 4452. Integral Extensions 4493. Characterizations of Dedekind Domains 4544. Ideals 4575. Finitely Generated Modules over Dedekind Domains 462Exercises 463Index 469