Exercises in Modules and Rings

Author: T.Y. Lam

Publisher: Springer Science & Business Media

ISBN: 9780387488998

Category: Mathematics

Page: 414

View: 9846

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.

Lectures on Modules and Rings

Author: Tsit-Yuen Lam

Publisher: Springer Science & Business Media

ISBN: 1461205255

Category: Mathematics

Page: 557

View: 8019

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.

Rings and Categories of Modules

Author: Frank W. Anderson,Kent R. Fuller

Publisher: Springer Science & Business Media

ISBN: 9780387978451

Category: Mathematics

Page: 376

View: 5488

This book is intended to provide a self-contained account of much of the theory of rings and modules. The theme of the text throughout is the relationship between the one-sided ideal structure a ring may possess and the behavior of its categories of modules. Following a brief outline of the foundations, the book begins with the basic definitions and properties of rings, modules and homomorphisms. The remainder of the text gives comprehensive treatments of direct sums, finiteness conditions, the Wedderburn-Artin Theorem, the Jacobson radical, the hom and tensor functions, Morita equivalence and duality, decomposition theory, and semiperfect and perfect rings. This second edition includes a chapter containing many of the classical results on Artinian rings that have helped form the foundation for much of contemporary research on the representation theory of Artinian rings and finite-dimensional algebras.

A First Course in Noncommutative Rings

Author: T.Y. Lam

Publisher: Springer Science & Business Media

ISBN: 1468404067

Category: Mathematics

Page: 397

View: 738

One of my favorite graduate courses at Berkeley is Math 251, a one-semester course in ring theory offered to second-year level graduate students. I taught this course in the Fall of 1983, and more recently in the Spring of 1990, both times focusing on the theory of noncommutative rings. This book is an outgrowth of my lectures in these two courses, and is intended for use by instructors and graduate students in a similar one-semester course in basic ring theory. Ring theory is a subject of central importance in algebra. Historically, some of the major discoveries in ring theory have helped shape the course of development of modern abstract algebra. Today, ring theory is a fer tile meeting ground for group theory (group rings), representation theory (modules), functional analysis (operator algebras), Lie theory (enveloping algebras), algebraic geometry (finitely generated algebras, differential op erators, invariant theory), arithmetic (orders, Brauer groups), universal algebra (varieties of rings), and homological algebra (cohomology of rings, projective modules, Grothendieck and higher K-groups). In view of these basic connections between ring theory and other branches of mathemat ics, it is perhaps no exaggeration to say that a course in ring theory is an indispensable part of the education for any fledgling algebraist. The purpose of my lectures was to give a general introduction to the theory of rings, building on what the students have learned from a stan dard first-year graduate course in abstract algebra.

Introductory Lectures on Rings and Modules

Author: John A. Beachy

Publisher: Cambridge University Press

ISBN: 9780521644075

Category: Mathematics

Page: 238

View: 8232

A first-year graduate text or reference for advanced undergraduates on noncommutative aspects of rings and modules.

Exercises in Classical Ring Theory

Author: T.Y. Lam

Publisher: Springer Science & Business Media

ISBN: 0387217711

Category: Mathematics

Page: 364

View: 9102

" This useful book, which grew out of the author's lectures at Berkeley, presents some 400 exercises of varying degrees of difficulty in classical ring theory, together with complete solutions, background information, historical commentary, bibliographic details, and indications of possible improvements or generalizations. The book should be especially helpful to graduate students as a model of the problem-solving process and an illustration of the applications of different theorems in ring theory. The author also discusses "the folklore of the subject: the 'tricks of the trade' in ring theory, which are well known to the experts in the field but may not be familiar to others, and for which there is usually no good reference". The problems are from the following areas: the Wedderburn-Artin theory of semisimple rings, the Jacobson radical, representation theory of groups and algebras, (semi)prime rings, (semi)primitive rings, division rings, ordered rings, (semi)local rings, the theory of idempotents, and (semi)perfect rings. Problems in the areas of module theory, category theory, and rings of quotients are not included, since they will appear in a later book. " (T. W. Hungerford, Mathematical Reviews)

Lectures on Rings and Modules

Author: Joachim Lambek

Publisher: American Mathematical Soc.

ISBN: 9780828422833

Category: Mathematics

Page: 183

View: 4892

Groups, Rings, Modules

Author: Maurice Auslander,David Buchsbaum

Publisher: Courier Corporation

ISBN: 048679542X

Category: Mathematics

Page: 480

View: 7974

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.

Integral Closure of Ideals, Rings, and Modules

Author: Craig Huneke,Irena Swanson

Publisher: Cambridge University Press

ISBN: 0521688604

Category: Mathematics

Page: 431

View: 6001

Ideal for graduate students and researchers, this book presents a unified treatment of the central notions of integral closure.

A Course in Commutative Algebra

Author: Gregor Kemper

Publisher: Springer Science & Business Media

ISBN: 9783642035456

Category: Mathematics

Page: 248

View: 883

This textbook offers a thorough, modern introduction into commutative algebra. It is intented mainly to serve as a guide for a course of one or two semesters, or for self-study. The carefully selected subject matter concentrates on the concepts and results at the center of the field. The book maintains a constant view on the natural geometric context, enabling the reader to gain a deeper understanding of the material. Although it emphasizes theory, three chapters are devoted to computational aspects. Many illustrative examples and exercises enrich the text.

Rational Homotopy Theory

Author: Yves Felix,Stephen Halperin,J.-C. Thomas

Publisher: Springer Science & Business Media

ISBN: 146130105X

Category: Mathematics

Page: 539

View: 3418

Rational homotopy theory is a subfield of algebraic topology. Written by three authorities in the field, this book contains all the main theorems of the field with complete proofs. As both notation and techniques of rational homotopy theory have been considerably simplified, the book presents modern elementary proofs for many results that were proven ten or fifteen years ago.

The Geometry of Schemes

Author: David Eisenbud,Joe Harris

Publisher: Springer Science & Business Media

ISBN: 0387226397

Category: Mathematics

Page: 300

View: 1945

Grothendieck’s beautiful theory of schemes permeates modern algebraic geometry and underlies its applications to number theory, physics, and applied mathematics. This simple account of that theory emphasizes and explains the universal geometric concepts behind the definitions. In the book, concepts are illustrated with fundamental examples, and explicit calculations show how the constructions of scheme theory are carried out in practice.

Cyclic Modules and the Structure of Rings

Author: S.K. Jain,Ashish K. Srivastava,Askar A. Tuganbaev

Publisher: Oxford University Press

ISBN: 019966451X

Category: Mathematics

Page: 220

View: 2169

This unique monograph brings together important material in the field of noncommutative rings and modules. It provides an up-to-date account of the topic of cyclic modules and the structure of rings which will be of particular interest to those working in abstract algebra and to graduate students who are exploring potential research topics.

Six Lectures on Commutative Algebra

Author: J. Elias,J. M. Giral,Rosa M. Miró-Roig,Santiago Zarzuela

Publisher: Springer Science & Business Media

ISBN: 3034603290

Category: Mathematics

Page: 402

View: 8573

Interest in commutative algebra has surged over the past decades. In order to survey and highlight recent developments in this rapidly expanding field, the Centre de Recerca Matematica in Bellaterra organized a ten-days Summer School on Commutative Algebra in 1996. Lectures were presented by six high-level specialists, L. Avramov (Purdue), M.K. Green (UCLA), C. Huneke (Purdue), P. Schenzel (Halle), G. Valla (Genova) and W.V. Vasconcelos (Rutgers), providing a fresh and extensive account of the results, techniques and problems of some of the most active areas of research. The present volume is a synthesis of the lectures given by these authors. Research workers as well as graduate students in commutative algebra and nearby areas will find a useful overview of the field and recent developments in it. Reviews "All six articles are at a very high level; they provide a thorough survey of results and methods in their subject areas, illustrated with algebraic or geometric examples." - Acta Scientiarum Mathematicarum Avramov lecture: "... it contains all the major results [on infinite free resolutions], it explains carefully all the different techniques that apply, it provides complete proofs (...). This will be extremely helpful for the novice as well as the experienced." - Mathematical reviews Huneke lecture: "The topic is tight closure, a theory developed by M. Hochster and the author which has in a short time proved to be a useful and powerful tool. (...) The paper is extremely well organized, written, and motivated." - Zentralblatt MATH Schenzel lecture: "... this paper is an excellent introduction to applications of local cohomology." - Zentralblatt MATH Valla lecture: "... since he is an acknowledged expert on Hilbert functions and since his interest has been so broad, he has done a superb job in giving the readers a lively picture of the theory." - Mathematical reviews Vasconcelos lecture: "This is a very useful survey on invariants of modules over noetherian rings, relations between them, and how to compute them." - Zentralblatt MATH

Introduction to Ring Theory

Author: Paul M. Cohn

Publisher: Springer Science & Business Media

ISBN: 1447104757

Category: Mathematics

Page: 229

View: 3298

A clear and structured introduction to the subject. After a chapter on the definition of rings and modules there are brief accounts of Artinian rings, commutative Noetherian rings and ring constructions, such as the direct product, Tensor product and rings of fractions, followed by a description of free rings. Readers are assumed to have a basic understanding of set theory, group theory and vector spaces. Over two hundred carefully selected exercises are included, most with outline solutions.

Introduction to Lie Algebras and Representation Theory

Author: J.E. Humphreys

Publisher: Springer Science & Business Media

ISBN: 1461263980

Category: Mathematics

Page: 173

View: 7409

This book is designed to introduce the reader to the theory of semisimple Lie algebras over an algebraically closed field of characteristic 0, with emphasis on representations. A good knowledge of linear algebra (including eigenvalues, bilinear forms, euclidean spaces, and tensor products of vector spaces) is presupposed, as well as some acquaintance with the methods of abstract algebra. The first four chapters might well be read by a bright undergraduate; however, the remaining three chapters are admittedly a little more demanding. Besides being useful in many parts of mathematics and physics, the theory of semisimple Lie algebras is inherently attractive, combining as it does a certain amount of depth and a satisfying degree of completeness in its basic results. Since Jacobson's book appeared a decade ago, improvements have been made even in the classical parts of the theory. I have tried to incor porate some of them here and to provide easier access to the subject for non-specialists. For the specialist, the following features should be noted: (I) The Jordan-Chevalley decomposition of linear transformations is emphasized, with "toral" subalgebras replacing the more traditional Cartan subalgebras in the semisimple case. (2) The conjugacy theorem for Cartan subalgebras is proved (following D. J. Winter and G. D. Mostow) by elementary Lie algebra methods, avoiding the use of algebraic geometry.

Fields and Rings

Author: Irving Kaplansky

Publisher: University of Chicago Press

ISBN: 9780226424514

Category: Mathematics

Page: 206

View: 2042

This book combines in one volume Irving Kaplansky's lecture notes on the theory of fields, ring theory, and homological dimensions of rings and modules. "In all three parts of this book the author lives up to his reputation as a first-rate mathematical stylist. Throughout the work the clarity and precision of the presentation is not only a source of constant pleasure but will enable the neophyte to master the material here presented with dispatch and ease."—A. Rosenberg, Mathematical Reviews

Rings, Modules, and Algebras in Stable Homotopy Theory

Author: Anthony D. Elmendorf,I. Kriz,M. A. Mandell,J. P. May

Publisher: American Mathematical Soc.

ISBN: 0821843036

Category: Mathematics

Page: 249

View: 1711

This book introduces a new point-set level approach to stable homotopy theory that has already had many applications and promises to have a lasting impact on the subject. Given the sphere spectrum $S$, the authors construct an associative, commutative, and unital smash product in a complete and cocomplete category of ``$S$-modules'' whose derived category is equivalent to the classical stable homotopy category. This construction allows for a simple and algebraically manageable definition of ``$S$-algebras'' and ``commutative $S$-algebras'' in terms of associative, or associative and commutative, products $R\wedge _SR \longrightarrow R$. These notions are essentially equivalent to the earlier notions of $A_{\infty }$ and $E_{\infty }$ ring spectra, and the older notions feed naturally into the new framework to provide plentiful examples. There is an equally simple definition of $R$-modules in terms of maps $R\wedge _SM\longrightarrow M$. When $R$ is commutative, the category of $R$-modules also has an associative, commutative, and unital smash product, and its derived category has properties just like the stable homotopy category. These constructions allow the importation into stable homotopy theory of a great deal of point-set level algebra.

Algebra I

Textbook for Students of Mathematics

Author: Alexey L. Gorodentsev

Publisher: Springer

ISBN: 3319452851

Category: Mathematics

Page: 564

View: 8960

This book is the first volume of an intensive “Russian-style” two-year graduate course in abstract algebra, and introduces readers to the basic algebraic structures – fields, rings, modules, algebras, groups, and categories – and explains the main principles of and methods for working with them. The course covers substantial areas of advanced combinatorics, geometry, linear and multilinear algebra, representation theory, category theory, commutative algebra, Galois theory, and algebraic geometry – topics that are often overlooked in standard undergraduate courses. This textbook is based on courses the author has conducted at the Independent University of Moscow and at the Faculty of Mathematics in the Higher School of Economics. The main content is complemented by a wealth of exercises for class discussion, some of which include comments and hints, as well as problems for independent study.