This book is an introduction to the theory of rings and modules that goes beyond what one normally obtains in a graduate course in abstract algebra. The theme of the text is the interplay between rings and modules. At times rings are investigated by considering given sets of conditions on the modules they admit and at other times rings of a certain type are considered to see what structure is forced on their modules. Standard topics in ring and module theory such as chain conditions on rings and modules, injective and projective modules and semisimple rings are included as well as subjects like category theory and homological algebra. The text also contains presentations on topics such as flat modules and coherent rings, injective envelopes, projective covers and perfect rings, reflexive modules and quasi-Frobenius rings, and graded rings and modules. The book is a self-contained volume written in a very systematic style: all proofs are clear and easy for the reader to understand and all arguments are based on materials contained in the book. A problem sets follow each section. It is assumed that the reader is familiar with concepts such as Zorn's lemma, commutative diagrams and ordinal and cardinal numbers. It is also assumed that the reader has a basic knowledge of rings and their homomorphisms. The text is suitable for graduate and PhD students who have chosen ring theory for their research subject. * Suitable for graduate students and researchers * Includes lots of exercises with tips Cover 1 Preface 6 About the Text 6 Acknowledgements 9 Contents 10 0 Preliminaries 16 0.1 Classes, Sets and Functions 16 Partial Orders and Equivalence Relations 17 Zorn’s Lemma and Well-Ordering 18 0.2 Ordinal and Cardinal Numbers 19 0.3 Commutative Diagrams 19 0.4 Notation and Terminology 19 Problem Set 20 1 Basic Properties of Rings and Modules 22 1.1 Rings 22 Problem Set 1.1 27 1.2 Left and Right Ideals 29 Factor Rings 32 Problem Set 1.2 32 1.3 Ring Homomorphisms 36 Problem Set 1.3 39 1.4 Modules 40 Factor Modules 44 Problem Set 1.4 45 1.5 Module Homomorphisms 47 Problem Set 1.5 51 2 Fundamental Constructions 54 2.1 Direct Products and Direct Sums 54 Direct Products 54 External Direct Sums 58 Internal Direct Sums 60 Problem Set 2.1 64 2.2 Free Modules 66 Rings with Invariant Basis Number 71 Problem Set 2.2 75 2.3 Tensor Products of Modules 78 Problem Set 2.3 84 3 Categories 86 3.1 Categories 86 Functors 90 Properties of Morphisms 92 Problem Set 3.1 94 3.2 Exact Sequences in ModR 97 Split Short Exact Sequences 99 Problem Set 3.2 101 3.3 Hom and as Functors 105 Properties of Hom 105 Properties of Tensor Products 109 Problem Set 3.3 110 3.4 Equivalent Categories and Adjoint Functors 112 Adjoints 114 Problem Set 3.4 117 4 Chain Conditions 119 4.1 Generating and Cogenerating Classes 119 Problem Set 4.1 121 4.2 Noetherian and Artinian Modules 122 Problem Set 4.2 133 4.3 Modules over Principal Ideal Domains 135 Free Modules over a PID 139 Finitely Generated Modules over a PID 143 Problem Set 4.3 149 5 Injective, Projective, and Flat Modules 150 5.1 Injective Modules 150 Injective Modules and the Functor HomR(—,M) 155 Problem Set 5.1 157 5.2 Projective Modules 159 Projective Modules and the Functor HomR (M, —) 163 Hereditary Rings 163 Semihereditary Rings 166 Problem Set 5.2 167 5.3 Flat Modules 169 Flat Modules and the Functor M R — 169 Coherent Rings 174 Regular Rings and Flat Modules 178 Problem Set 5.3 180 5.4 Quasi-Injective and Quasi-Projective Modules 184 Problem Set 5.4 185 6 Classical Ring Theory 186 6.1 The Jacobson Radical 186 Problem Set 6.1 192 6.2 The Prime Radical 193 Prime Rings 194 Semiprime Rings 199 Problem Set 6.2 201 6.3 Radicals and Chain Conditions 203 Problem Set 6.3 205 6.4 Wedderburn–Artin Theory 206 Problem Set 6.4 218 6.5 Primitive Rings and Density 220 Problem Set 6.5 225 6.6 Rings that Are Semisimple 226 Problem Set 6.6 230 7 Envelopes and Covers 231 7.1 Injective Envelopes 231 Problem Set 7.1 234 7.2 Projective Covers 236 The Radical of a Projective Module 238 Semiperfect Rings 242 Perfect Rings 252 Problem Set 7.2 256 7.3 QI-Envelopes and QP-Covers 260 Quasi-Injective Envelopes 260 Quasi-Projective Covers 263 Problem Set 7.3 267 8 Rings and Modules of Quotients 269 8.1 Rings of Quotients 269 The Noncommutative Case 269 The Commutative Case 276 Problem Set 8.1 277 8.2 Modules of Quotients 279 Problem Set 8.2 282 8.3 Goldie’s Theorem 285 Problem Set 8.3 291 8.4 The Maximal Ring of Quotients 292 Problem Set 8.4 306 9 Graded Rings and Modules 309 9.1 Graded Rings and Modules 309 Graded Rings 309 Graded Modules 314 Problem Set 9.1 320 9.2 Fundamental Concepts 323 Graded Direct Products and Sums 324 Graded Tensor Products 326 Graded Free Modules 327 Problem Set 9.2 328 9.3 Graded Projective, Graded Injective and Graded Flat Modules 329 Graded Projective and Graded Injective Modules 329 Graded Flat Modules 333 Problem Set 9.3 334 9.4 Graded Modules with Chain Conditions 335 Graded Noetherian and Graded Artinian Modules 335 Problem Set 9.4 340 9.5 More on Graded Rings 340 The Graded Jacobson Radical 340 Graded Wedderburn–Artin Theory 342 Problem Set 9.5 344 10 More on Rings and Modules 346 10.1 Reflexivity and Vector Spaces 347 Problem Set 10.1 349 10.2 Reflexivity and R-modules 350 Self-injective Rings 351 Kasch Rings and Injective Cogenerators 354 Semiprimary Rings 356 Quasi-Frobenius Rings 357 Problem Set 10.2 361 11 Introduction to Homological Algebra 363 11.1 Chain and Cochain Complexes 363 Homology and Cohomology Sequences 368 Problem Set 11.1 372 11.2 Projective and Injective Resolutions 374 Problem Set 11.2 380 11.3 Derived Functors 382 Problem Set 11.3 387 11.4 Extension Functors 388 Right Derived Functors of HomR(—, X) 388 Right Derived Functors of HomR(X, —) 396 Problem Set 11.4 401 11.5 Torsion Functors 405 Left Derived Functors of — r X and X R — 405 Problem Set 11.5 409 12 Homological Methods 410 12.1 Projective and Injective Dimension 410 Problem Set 12.1 417 12.2 Flat Dimension 418 Problem Set 12.2 422 12.3 Dimension of Polynomial Rings 424 Problem Set 12.3 432 12.4 Dimension of Matrix Rings 432 Problem Set 12.4 436 12.5 Quasi-Frobenius Rings Revisited 436 More on Reflexive Modules 436 Problem Set 12.5 443 A Ordinal and Cardinal Numbers 444 Ordinal Numbers 444 Cardinal Numbers 448 Problem Set 450 Bibliography 452 List of Symbols 456 Index 458
This book is an introduction to the theory of rings and modules that goes beyond what one normally obtains in a graduate course in abstract algebra. The theme of the text is the interplay between rings and modules. At times rings are investigated by considering given sets of conditions on the modules they admit and at other times rings of a certain type are considered to see what structure is forced on their modules. Standard topics in ring and module theory such as chain conditions on rings and modules, injective and projective modules and semisimple rings are included as well as subjects like category theory and homological algebra. The text also contains presentations on topics such as flat modules and coherent rings, injective envelopes, projective covers and perfect rings, reflexive modules and quasi-Frobenius rings, and graded rings and modules.
The book is a self-contained volume written in a very systematic style: all proofs are clear and easy for the reader to understand and all arguments are based on materials contained in the book. A problem sets follow each section.
It is assumed that the reader is familiar with concepts such as Zorn's lemma, commutative diagrams and ordinal and cardinal numbers. It is also assumed that the reader has a basic knowledge of rings and their homomorphisms. The text is suitable for graduate and PhD students who have chosen ring theory for their research subject.
This book is an introduction to the theory of rings and modules that goes beyond what one normally obtains in a graduate course in abstract algebra. In addition to the presentation of standard topics in ring and module theory, it also covers category theory, homological algebra and even more specialized topics like injective envelopes and projective covers, reflexive modules and quasi-Frobenius rings, and graded rings and modules. The book is a self-contained volume written in a very systematic style: all proofs are clear and easy for the reader to understand and all arguments are based on materials contained in the book. A problem sets follow each section. It is suitable for graduate and PhD students who have chosen ring theory for their research subject.
