Rings And Categories Of Modules

Author : Frank W. Anderson,Kent R. Fuller
Publisher : Springer Science & Business Media
Page : 386 pages
File Size : 50,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461244189

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Rings and Categories of Modules by Frank W. Anderson,Kent R. Fuller Pdf

This book is intended to provide a reasonably self-contained account of a major portion of the general theory of rings and modules suitable as a text for introductory and more advanced graduate courses. We assume the famil iarity with rings usually acquired in standard undergraduate algebra courses. Our general approach is categorical rather than arithmetical. The continuing theme of the text is the study of the relationship between the one-sided ideal structure that a ring may possess and the behavior of its categories of modules. Following a brief outline of set-theoretic and categorical foundations, the text begins with the basic definitions and properties of rings, modules and homomorphisms and ranges through comprehensive treatments of direct sums, finiteness conditions, the Wedderburn-Artin Theorem, the Jacobson radical, the hom and tensor functions, Morita equivalence and duality, de composition theory of injective and projective modules, and semi perfect and perfect rings. In this second edition we have included a chapter containing many of the classical results on artinian rings that have hdped to form the foundation for much of the contemporary research on the representation theory of artinian rings and finite dimensional algebras. Both to illustrate the text and to extend it we have included a substantial number of exercises covering a wide spectrum of difficulty. There are, of course" many important areas of ring and module theory that the text does not touch upon.

Author : F.W. Anderson,K.R. Fuller
Publisher : Springer Science & Business Media
Page : 346 pages
File Size : 47,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781468499131

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Rings and Categories of Modules by F.W. Anderson,K.R. Fuller Pdf

This book is intended to provide a reasonably self-contained account of a major portion of the general theory of rings and modules suitable as a text for introductory and more advanced graduate courses. We assume the famil iarity with rings usually acquired in standard undergraduate algebra courses. Our general approach is categorical rather than arithmetical. The continuing theme of the text is the study of the relationship between the one-sided ideal structure that a ring may possess and the behavior of its categories of modules. Following a brief outline of set-theoretic and categorical foundations, the text begins with the basic definitions and properties of rings, modules and homomorphisms and ranges through comprehensive treatments of direct sums, finiteness conditions, the Wedderburn-Art in Theorem, the Jacobson radical, the hom and tensor functions, Morita equivalence and duality, de composition theory of injective and projective modules, and semiperfect and perfect rings. Both to illustrate the text and to extend it we have included a substantial number of exercises covering a wide spectrum of difficulty. There are, of course, many important areas of ring and module theory that the text does not touch upon. For example, we have made no attempt to cover such subjects as homology, rings of quotients, or commutative ring theory.

Author : Tsit-Yuen Lam
Publisher : Springer Science & Business Media
Page : 577 pages
File Size : 49,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461205258

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Lectures on Modules and Rings by Tsit-Yuen Lam Pdf

This new book can be read independently from the first volume and may be used for lecturing, seminar- and self-study, or for general reference. It focuses more on specific topics in order to introduce readers to a wealth of basic and useful ideas without the hindrance of heavy machinery or undue abstractions. User-friendly with its abundance of examples illustrating the theory at virtually every step, the volume contains a large number of carefully chosen exercises to provide newcomers with practice, while offering a rich additional source of information to experts. A direct approach is used in order to present the material in an efficient and economic way, thereby introducing readers to a considerable amount of interesting ring theory without being dragged through endless preparatory material.

Author : Maurice Auslander,David Buchsbaum
Publisher : Courier Corporation
Page : 484 pages
File Size : 43,8 Mb
Release : 2014-06-01
Category : Mathematics
ISBN : 9780486795423

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Groups, Rings, Modules by Maurice Auslander,David Buchsbaum Pdf

Classic monograph covers sets and maps, monoids and groups, unique factorization domains, localization and tensor products, applications of fundamental theorem, algebraic field extension, Dedekind domains, and much more. 1974 edition.

Author : John Dauns
Publisher : Cambridge University Press
Page : 470 pages
File Size : 55,8 Mb
Release : 1994-10-28
Category : Mathematics
ISBN : 9780521462587

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Modules and Rings by John Dauns Pdf

This 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.

Author : Robert Wisbauer
Publisher : Routledge
Page : 425 pages
File Size : 51,8 Mb
Release : 2018-05-11
Category : Mathematics
ISBN : 9781351447348

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Foundations of Module and Ring Theory by Robert Wisbauer Pdf

This volume provides a comprehensive introduction to module theory and the related part of ring theory, including original results as well as the most recent work. It is a useful and stimulating study for those new to the subject as well as for researchers and serves as a reference volume. Starting form a basic understanding of linear algebra, the theory is presented and accompanied by complete proofs. For a module M, the smallest Grothendieck category containing it is denoted by o[M] and module theory is developed in this category. Developing the techniques in o[M] is no more complicated than in full module categories and the higher generality yields significant advantages: for example, module theory may be developed for rings without units and also for non-associative rings. Numerous exercises are included in this volume to give further insight into the topics covered and to draw attention to related results in the literature.

Author : Frank W Anderson,Kent R Fuller
Publisher : Unknown
Page : 392 pages
File Size : 43,6 Mb
Release : 1998-08-18
Category : Electronic
ISBN : 1461244196

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Rings and Categories of Modules by Frank W Anderson,Kent R Fuller Pdf

Author : A. J. Berrick,M. E. Keating
Publisher : Cambridge University Press
Page : 286 pages
File Size : 41,9 Mb
Release : 2000-05
Category : Mathematics
ISBN : 0521632749

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An Introduction to Rings and Modules by A. J. Berrick,M. E. Keating Pdf

This is a concise 2000 introduction at graduate level to ring theory, module theory and number theory.

Author : Carl Faith
Publisher : Springer Science & Business Media
Page : 589 pages
File Size : 43,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642806346

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Algebra by Carl Faith Pdf

VI of Oregon lectures in 1962, Bass gave simplified proofs of a number of "Morita Theorems", incorporating ideas of Chase and Schanuel. One of the Morita theorems characterizes when there is an equivalence of categories mod-A R::! mod-B for two rings A and B. Morita's solution organizes ideas so efficiently that the classical Wedderburn-Artin theorem is a simple consequence, and moreover, a similarity class [AJ in the Brauer group Br(k) of Azumaya algebras over a commutative ring k consists of all algebras B such that the corresponding categories mod-A and mod-B consisting of k-linear morphisms are equivalent by a k-linear functor. (For fields, Br(k) consists of similarity classes of simple central algebras, and for arbitrary commutative k, this is subsumed under the Azumaya [51]1 and Auslander-Goldman [60J Brauer group. ) Numerous other instances of a wedding of ring theory and category (albeit a shot gun wedding!) are contained in the text. Furthermore, in. my attempt to further simplify proofs, notably to eliminate the need for tensor products in Bass's exposition, I uncovered a vein of ideas and new theorems lying wholely within ring theory. This constitutes much of Chapter 4 -the Morita theorem is Theorem 4. 29-and the basis for it is a corre spondence theorem for projective modules (Theorem 4. 7) suggested by the Morita context. As a by-product, this provides foundation for a rather complete theory of simple Noetherian rings-but more about this in the introduction.

Author : Alberto Facchini,Evan Houston,Luigi Salce
Publisher : CRC Press
Page : 530 pages
File Size : 45,5 Mb
Release : 2020-02-10
Category : Mathematics
ISBN : 0824750810

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Rings, Modules, Algebras, and Abelian Groups by Alberto Facchini,Evan Houston,Luigi Salce Pdf

Rings, Modules, Algebras, and Abelian Groups summarizes the proceedings of a recent algebraic conference held at Venice International University in Italy. Surveying the most influential developments in the field, this reference reviews the latest research on Abelian groups, algebras and their representations, module and ring theory, and topological

Author : A. J. Berrick,M. E. Keating
Publisher : Cambridge University Press
Page : 384 pages
File Size : 55,5 Mb
Release : 2000-05-25
Category : Mathematics
ISBN : 0521632765

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Categories and Modules with K-Theory in View by A. J. Berrick,M. E. Keating Pdf

This book, first published in 2000, is a concise introduction at graduate level to ring theory, module theory and number theory.

Author : T.Y. Lam
Publisher : Springer Science & Business Media
Page : 427 pages
File Size : 55,9 Mb
Release : 2009-12-08
Category : Mathematics
ISBN : 9780387488998

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Exercises in Modules and Rings by T.Y. Lam Pdf

This volume offers a compendium of exercises of varying degree of difficulty in the theory of modules and rings. It is the companion volume to GTM 189. All exercises are solved in full detail. Each section begins with an introduction giving the general background and the theoretical basis for the problems that follow.

Author : Paul E. Bland
Publisher : Walter de Gruyter
Page : 467 pages
File Size : 44,8 Mb
Release : 2011
Category : Mathematics
ISBN : 9783110250220

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Rings and Their Modules by Paul E. Bland Pdf

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 proj

Author : Alberto Facchini
Publisher : Springer Science & Business Media
Page : 296 pages
File Size : 40,7 Mb
Release : 2012-02-03
Category : Mathematics
ISBN : 9783034803038

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Module Theory by Alberto Facchini Pdf

This book presents topics in module theory and ring theory: some, such as Goldie dimension and semiperfect rings are now considered classical and others more specialized, such as dual Goldie dimension, semilocal endomorphism rings, serial rings and modules.

Author : Frank Wylie Anderson,Kent R. Fuller
Publisher : Unknown
Page : 402 pages
File Size : 43,9 Mb
Release : 1992
Category : Mathematics
ISBN : UOM:39015029192914

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Rings and Categories of Modules by Frank Wylie Anderson,Kent R. Fuller Pdf