Modules and RingsThis book on modern module and non-commutative ring theory is ideal for beginning graduate students. It starts at the foundations of the subject and progresses rapidly through the basic concepts to help the reader reach current research frontiers. Students will have the chance to develop proofs, solve problems, and to find interesting questions. The first half of the book is concerned with free, projective, and injective modules, tensor algebras, simple modules and primitive rings, the Jacobson radical, and subdirect products. Later in the book, more advanced topics, such as hereditary rings, categories and functors, flat modules, and purity are introduced. These later chapters will also prove a useful reference for researchers in non-commutative ring theory. Enough background material (including detailed proofs) is supplied to give the student a firm grounding in the subject. |
Contents
MODULES | 1 |
FREE MODULES | 19 |
INJECTIVE MODULES | 30 |
TENSOR PRODUCTS | 53 |
CERTAIN IMPORTANT ALGEBRAS | 71 |
SIMPLE MODULES | 86 |
THE JACOBSON RADICAL | 111 |
SUBDIRECT PRODUCT | 140 |
HEREDITARY RINGS FREE | 269 |
MODULE CONSTRUCTIONS | 283 |
CATEGORIES AND FUNCTORS | 298 |
MODULE CATEGORIES | 335 |
FLAT MODULES | 359 |
PURITY | 367 |
APPENDIX A BASICS | 398 |
APPENDIX B CERTAIN IMPORTANT ALGEBRAS | 412 |
PRIMES AND SEMIPRIMES | 148 |
PROJECTIVE MODULES AND MORE | 163 |
53 | 172 |
DIRECT SUM DECOMPOSITIONS | 204 |
SIMPLE ALGEBRAS | 239 |
LIST OF SYMBOLS AND NOTATION | 427 |
| 433 | |
| 436 | |
| 442 | |
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Common terms and phrases
a₁ abelian group arbitrary Artinian assume b₁ C₁ chapter commutative diagram commutative ring contains Corollary defined Definition denote direct limit direct sum direct summand division ring e₁ equivalence class exterior algebras F-algebra F₁ finite number free module functor h₁ Hence hypothesis identity element indecomposable index set injective hull injective module integer isomorphism Jacobson radical kernel L₁ left R-module Lemma M₁ matrix maximal right ideal minimal monic Morph morphism nonassociative ring nonzero notation ordinal P₁ phism primitive primitive ring projective module Proof Proposition Prove pure essential extension pure injective quasi-regular quotient R-map R₁ right R-module S₁ satisfies semiprime short exact sequence solution x₁ subdirect product submodule subring subset Suppose system of equations tensor product theorem U₁ universal property V₁ vector space Xirij y₁ zero π₁
Popular passages
Page x - Simiarly, the chapter on module categories and the chapter on flat modules develop quickly some of the more important facts, and do not cover these subjects exhaustively. The last chapter on systems of equations in modules, pure projectivity, pure injectivity, and pure injective hulls is somewhat more advanced. At this point in time, the author does not know of any textbook in print which develops this subject logically from the beginning as is done here.
Page ix - This book does not dwell too long on any one topic and thus is suitable for courses where a wide range of topics have to be covered quickly. This is also the reason why the chapters on category theory, functors, module categories, and more complicated facts about tensor products are at the end of the book.
Page viii - This text has more material than can be covered in a one year course. Although there are no real prerequisites aside from linear algebra, an introductory abstract algebra course, which usually includes groups, fields, and some commutative rings, might be helpful.
Page x - Abelian categories would suffice for applications to modules, nevertheless category theory is covered in greater generality so as to make it also applicable to other fields, such as topology or partially ordered systems, where greater generality is required.