Foundations Of Module And Ring Theory

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Foundations of Module and Ring Theory

Robert Wisbauer
Foundations of Module and Ring Theory Book PDF

Author : Robert Wisbauer
Publisher : Routledge
Page : 606 pages
File Size : 41,7 Mb
Release : 2018-05-11
Category : Mathematics
ISBN : 9781351447355

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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.

Foundations of Module and Ring Theory

Robert Wisbauer
Foundations of Module and Ring Theory Book PDF

Author : Robert Wisbauer
Publisher : Routledge
Page : 425 pages
File Size : 55,8 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.

Modules and Rings

John Dauns
Modules and Rings Book PDF

Author : John Dauns
Publisher : Cambridge University Press
Page : 470 pages
File Size : 52,6 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.

Foundations of Commutative Rings and Their Modules

Fanggui Wang,Hwankoo Kim
Foundations of Commutative Rings and Their Modules Book PDF

Author : Fanggui Wang,Hwankoo Kim
Publisher : Springer
Page : 699 pages
File Size : 50,7 Mb
Release : 2017-01-06
Category : Mathematics
ISBN : 9789811033377

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Foundations of Commutative Rings and Their Modules by Fanggui Wang,Hwankoo Kim Pdf

This book provides an introduction to the basics and recent developments of commutative algebra. A glance at the contents of the first five chapters shows that the topics covered are ones that usually are included in any commutative algebra text. However, the contents of this book differ significantly from most commutative algebra texts: namely, its treatment of the Dedekind–Mertens formula, the (small) finitistic dimension of a ring, Gorenstein rings, valuation overrings and the valuative dimension, and Nagata rings. Going further, Chapter 6 presents w-modules over commutative rings as they can be most commonly used by torsion theory and multiplicative ideal theory. Chapter 7 deals with multiplicative ideal theory over integral domains. Chapter 8 collects various results of the pullbacks, especially Milnor squares and D+M constructions, which are probably the most important example-generating machines. In Chapter 9, coherent rings with finite weak global dimensions are probed, and the local ring of weak global dimension two is elaborated on by combining homological tricks and methods of star operation theory. Chapter 10 is devoted to the Grothendieck group of a commutative ring. In particular, the Bass–Quillen problem is discussed. Finally, Chapter 11 aims to introduce relative homological algebra, especially where the related concepts of integral domains which appear in classical ideal theory are defined and investigated by using the class of Gorenstein projective modules. Each section of the book is followed by a selection of exercises of varying degrees of difficulty. This book will appeal to a wide readership from graduate students to academic researchers who are interested in studying commutative algebra.

Rings and Categories of Modules

Frank W. Anderson,Kent R. Fuller
Rings and Categories of Modules Book PDF

Author : Frank W. Anderson,Kent R. Fuller
Publisher : Springer Science & Business Media
Page : 386 pages
File Size : 43,7 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.

Modules and the Structure of Rings

Golan
Modules and the Structure of Rings Book PDF

Author : Golan
Publisher : CRC Press
Page : 272 pages
File Size : 45,7 Mb
Release : 2017-10-19
Category : Mathematics
ISBN : 9781351430371

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Modules and the Structure of Rings by Golan Pdf

This textbook is designed for students with at least one solid semester of abstract algebra,some linear algebra background, and no previous knowledge of module theory. Modulesand the Structure of Rings details the use of modules over a ring as a means of consideringthe structure of the ring itself--explaining the mathematics and "inductivereasoning" used in working on ring theory challenges and emphasizing modules insteadof rings.Stressing the inductive aspect of mathematical research underlying the formal deductivestyle of the literature, this volume offers vital background on current methods for solvinghard classification problems of algebraic structures. Written in an informal butcompletely rigorous style, Modules and the Structure of Rings clarifies sophisticatedproofs ... avoids the formalism of category theory ... aids independent study or seminarwork ... and supplies end-of-chapter problems.This book serves as an excellent primary.text for upper-level undergraduate and graduatestudents in one-semester courses on ring or module theory-laying a foundation formore advanced study of homological algebra or module theory.

Ring and Module Theory

Toma Albu,Gary F. Birkenmeier,Ali Erdogan,Adnan Tercan
Ring and Module Theory Book PDF

Author : Toma Albu,Gary F. Birkenmeier,Ali Erdogan,Adnan Tercan
Publisher : Springer Science & Business Media
Page : 200 pages
File Size : 46,5 Mb
Release : 2011-02-04
Category : Mathematics
ISBN : 9783034600071

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Ring and Module Theory by Toma Albu,Gary F. Birkenmeier,Ali Erdogan,Adnan Tercan Pdf

This book is a collection of invited papers and articles, many presented at the 2008 International Conference on Ring and Module Theory. The papers explore the latest in various areas of algebra, including ring theory, module theory and commutative algebra.

Algebra

Carl Faith
Algebra Book PDF

Author : Carl Faith
Publisher : Springer Science & Business Media
Page : 589 pages
File Size : 46,9 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.

Algebras, Rings and Modules

Michiel Hazewinkel,Nadiya Gubareni,V.V. Kirichenko
Algebras Rings and Modules Book PDF

Author : Michiel Hazewinkel,Nadiya Gubareni,V.V. Kirichenko
Publisher : Springer Science & Business Media
Page : 393 pages
File Size : 49,9 Mb
Release : 2006-01-18
Category : Mathematics
ISBN : 9781402026911

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Algebras, Rings and Modules by Michiel Hazewinkel,Nadiya Gubareni,V.V. Kirichenko Pdf

Accosiative rings and algebras are very interesting algebraic structures. In a strict sense, the theory of algebras (in particular, noncommutative algebras) originated fromasingleexample,namelythequaternions,createdbySirWilliamR.Hamilton in1843. Thiswasthe?rstexampleofanoncommutative”numbersystem”. During thenextfortyyearsmathematiciansintroducedotherexamplesofnoncommutative algebras, began to bring some order into them and to single out certain types of algebras for special attention. Thus, low-dimensional algebras, division algebras, and commutative algebras, were classi?ed and characterized. The ?rst complete results in the structure theory of associative algebras over the real and complex ?elds were obtained by T.Molien, E.Cartan and G.Frobenius. Modern ring theory began when J.H.Wedderburn proved his celebrated cl- si?cation theorem for ?nite dimensional semisimple algebras over arbitrary ?elds. Twenty years later, E.Artin proved a structure theorem for rings satisfying both the ascending and descending chain condition which generalized Wedderburn structure theorem. The Wedderburn-Artin theorem has since become a corn- stone of noncommutative ring theory. The purpose of this book is to introduce the subject of the structure theory of associative rings. This book is addressed to a reader who wishes to learn this topic from the beginning to research level. We have tried to write a self-contained book which is intended to be a modern textbook on the structure theory of associative rings and related structures and will be accessible for independent study.

Groups, Rings, Modules

Maurice Auslander,David Buchsbaum
Groups Rings Modules Book PDF

Author : Maurice Auslander,David Buchsbaum
Publisher : Courier Corporation
Page : 484 pages
File Size : 44,7 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.

An Introduction to Rings and Modules

A. J. Berrick,M. E. Keating
An Introduction to Rings and Modules Book PDF

Author : A. J. Berrick,M. E. Keating
Publisher : Cambridge University Press
Page : 286 pages
File Size : 44,9 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.

Modules and the Structure of Rings

Golan
Modules and the Structure of Rings Book PDF

Author : Golan
Publisher : CRC Press
Page : 298 pages
File Size : 54,8 Mb
Release : 1991-04-24
Category : Mathematics
ISBN : 082478555X

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Modules and the Structure of Rings by Golan Pdf

This book offers vital background information on methods for solving hard classification problems of algebraic structures. It explains how algebraists deal with the problem of the structure of modules over rings and how they make use of these structures to classify rings.

Classes of Modules

John Dauns,Yiqiang Zhou
Classes of Modules Book PDF

Author : John Dauns,Yiqiang Zhou
Publisher : CRC Press
Page : 232 pages
File Size : 43,7 Mb
Release : 2006-06-19
Category : Mathematics
ISBN : 9781420011593

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Classes of Modules by John Dauns,Yiqiang Zhou Pdf

Because traditional ring theory places restrictive hypotheses on all submodules of a module, its results apply only to small classes of already well understood examples. Often, modules with infinite Goldie dimension have finite-type dimension, making them amenable to use with type dimension, but not Goldie dimension. By working with natural classes

Rings and Their Modules

Paul E. Bland
Rings and Their Modules Book PDF

Author : Paul E. Bland
Publisher : Walter de Gruyter
Page : 467 pages
File Size : 46,9 Mb
Release : 2011-02-01
Category : Mathematics
ISBN : 9783110250237

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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. 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.

Exercises in Modules and Rings

T.Y. Lam
Exercises in Modules and Rings Book PDF

Author : T.Y. Lam
Publisher : Springer Science & Business Media
Page : 427 pages
File Size : 54,8 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.