Groups, Rings, Modules

The main thrust of this book is easily described. It is to introduce the reader who already has some familiarity with the basic notions of sets, groups, rings, and vector spaces to the study of rings by means of their module theory. This program is carried out in a systematic way for the classicalJy important semisimple rings, principal ideal domains, and Oedekind domains. The proofs of the well-known basic properties of these traditionally important rings have been designed to emphasize general concepts and techniques. HopefulJy this wilJ give the reader a good introduction to the unifying methods currently being developed in ring theory. CONTENTS Preface ix PART ONE 1 Chapter I SETS AND MAPS 3 I. Sets and Subsets 3 2. Maps S 3. Isomorphisms of Sets 7 4. Epimorphisms and Monomorphisms 8 S. The Image Analysis of a Map 10 6. The Coimage Analysis of a Map II 7. Description of Surjective Maps 12 8. Equivalence Relations 13 9. Cardinality of Sets IS 10. Ordered Sets 16 II. Axiom of Choice 17 12. Products and Sums of Sets 20 Exercises 23 Chapter 2 MONOIDS AND GROUPS 27 1. Monoids 27 2. Morphisms of Monoids 30 3. Special Types of Morphisms 32 4. Analyses of Morphisms 37 5. Description of Surjective Morphisms 39 6. Groups and Morphisms of Groups 41 7. Kernels of Morphisms of Groups 43 8. Groups of Fractions 49 9. The Integers 55 10. Finite and Infinite Sets 57 Exercises 64 Chapter 3 CATEGORIES 75 1. Categories 75 2. Morphisms 79 3. Products and Sums 82 Exercises 85 Chapter 4 RINGS 99 1. Category of Rings 99 2. Polynomial Rings 103 3. Analyses of Ring Morphisms 107 4. Ideals 112 5. Products of Rings 115 Exercises 116 PART TWO 127 Chapter 5 UNIQUE FACTORIZATION DOMAINS 129 I. Divisibility 130 2. Integral Domains 133 3. Unique Factorization Domains 138 4. Divisibility in UFD's 140 5. Principal Ideal Domains 147 6. Factor Rings of PID's 152 7. Divisors 155 8. Localization in Integral Domains 159 9. A Criterion for Unique Factorization 164 10. When R [X] is a UFD 169 Exercises 171 Chapter 6 GENERAL MODULE THEORY 176 1. Category of Modules over a Ring 178 2. The Composition Maps in Mod(R) 183 3. Analyses of R-Module Morphisms 185 4. Exact Sequences 193 5. Isomorphism Theorems 201 6. Noetherian and Artinian Modules 206 7. Free R-Modules 210 8. Characterization of Division Rings 216 9. Rank of Free Modules 221 10. Complementary Submodules of a Module 224 11. Sums of Modules 231 CONTENTS vII 12. Change of Rings 239 13. Torsion Modules over PID's 242 14. Products of Modules 246 Exercises 248 Chapter 7 SEMISIMPLE RINGS AND MODULES 266 I. Simple Rings 266 2. Semisimple Modules 271 3. Projective Modules 276 4. The Opposite Ring 280 Exercises 283 Chapter 8 ARTINIAN RINGS 289 1. Idempotents in Left Artinian Rings 289 2. The Radical of a Left Artinian Ring 294 3. The Radical of an Arbitrary Ring 298 Exercises 302 PART THREE 311 Chapter 9 LOCALIZATION AND TENSOR PRODUCTS 313 1. Localization of Rings 313 2. Localization of Modules 316 3. Applications of Localization 320 4. Tensor Products 323 5. Morphisms of Tensor Products 328 6. Locally Free Modules 334 Exercises 337 Chapter 10 PRINCIPAL IDEAL DOMAINS 351 I. Submodules of Free Modules 352 2. Free Submodules of Free Modules 355 3. Finitely Generated Modules over PID's 359 4. Injective Modules 363 5. The Fundamental Theorem for PID's 366 Exercises 371 Chapter II APPLICATIONS OF FUNDAMENTAL THEOREM 376 I. Diagonalization 376 2. Determinants 380 3. Mat rices 387 4. Further Applications of the Fundamental Theorem 391 5. Canonical Forms 395 Exercises 40 I PART FOUR 413 Chapter 12 ALGEBRAIC FIELD EXTENSIONS 415 1. Roots of Polynomials 415 2. Algebraic Elements 420 3. Morphisms of Fields 425 4. Separability 430 5. Galois Extensions 434 Exercises 440 Chapter 13 DEDEKIND DOMAINS 445 I. Dedekind Domains 445 2. Integral Extensions 449 3. Characterizations of Dedekind Domains 454 4. Ideals 457 5. Finitely Generated Modules over Dedekind Domains 462 Exercises 463 Index 469