Exercises in Classical Ring Theory

Exercises in Classical Ring Theory

‘ 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. ‘

Exercises in Classical Ring Theory

The Mathematics of Logic

The Mathematics of Logic

This undergraduate textbook covers the key material for a typical first course in logic, in particular presenting a full mathematical account of the most important result in logic, the Completeness Theorem for first-order logic. Looking at a series of interesting systems, increasing in complexity, then proving and discussing the Completeness Theorem for each, the author ensures that the number of new concepts to be absorbed at each stage is manageable, whilst providing lively mathematical applications throughout. Unfamiliar terminology is kept to a minimum, no background in formal set-theory is required, and the book contains proofs of all the required set theoretical results. The reader is taken on a journey starting with Knig’s Lemma, and progressing via order relations, Zorn’s Lemma, Boolean algebras, and propositional logic, to completeness and compactness of first-order logic. As applications of the work on first-order logic, two final chapters provide introductions to model theory and nonstandard analysis.

The Mathematics of Logic

Recent Advances in Operator Theory, Operator Algebras and Their Applications

Recent Advances in Operator Theory, Operator Algebras and Their Applications

This book offers peer-reviewed articles from the 19th International Conference on Operator Theory, Summer 2002. It contains recent developments in a broad range of topics from operator theory, operator algebras and their applications, particularly to differential analysis, complex functions, ergodic theory, mathematical physics, matrix analysis, and systems theory. The book covers a large variety of topics including single operator theory, C*-algebras, diffrential operators, integral transforms, stochastic processes and operators, and more.

Recent Advances in Operator Theory, Operator Algebras and Their Applications

Division Algebras, Lattices, Physics, Windmill Tilting

Division Algebras, Lattices, Physics, Windmill Tilting

Complex numbers, quaternions, octonions, integral domains, laminated lattices, the Leech lattice, with applications to theoretical physics, including the Standard Model of quarks and leptons with spinors and gauge groups.

Division Algebras, Lattices, Physics, Windmill Tilting

The W3 Algebra

The W3 Algebra

‘W’ algebras are nonlinear generalizations of Lie algebras that arise in the context of two-dimensional conformal field theories when one explores higher-spin extensions of the Virasoro algebra. They provide the underlying symmetry algebra of certain string generalizations which allow the extended world sheet gravity. This book presents such gravity theories, concentrating on the algebra of physical operators determined from an analysis of the corresponding BRST cohomology. It develops the representation theory of ‘W’ algebras needed to extend the standard techniques which were so successful in treating linear algebras. For certain strings corresponding to ‘W’N gravity we show that the operator cohomology has a natural geometric model. This result suggests new directions for the study of ‘W’ geometry.

The W3 Algebra

Lectures on Infinite-Dimensional Lie Algebra

Lectures on Infinite-Dimensional Lie Algebra

The representation theory of affine Lie algebras has been developed in close connection with various areas of mathematics and mathematical physics in the last two decades. There are three excellent books on it, written by Victor G Kac. This book begins with a survey and review of the material treated in Kac’s books. In particular, modular invariance and conformal invariance and are explained in more detail. The book then goes further, dealing with some of the recent topics involving the representation theory of affine Lie algebras. Since these topics are important not only in themselves but also in their application to some areas of mathematics and mathematical physics, the book expounds them with examples and detailed calculations.

Lectures on Infinite-Dimensional Lie Algebra

Basic Theory of Algebraic Groups and Lie Algebras

Basic Theory of Algebraic Groups and Lie Algebras

he theory of algebraic groups results from the interaction of various basic techniques from field theory, multilinear algebra, commutative ring theory, algebraic geometry and general algebraic representation theory of groups and

Basic Theory of Algebraic Groups and Lie Algebras

Noncommutative Noetherian rings

Noncommutative Noetherian rings

Provides a comprehensive account of the major developments in this important branch of ring theory which have taken place over the past 30 years. Much of the material which comprises this volume has not appeared anywhere in book form before, and the authors have improved and simplified many of the accounts available in journals. The first few chapters form a basic course” which introduces the reader to the subject. Subsequent chapters are each relatively self-contained, so that readers interested in a particular subject can easily consult the sections they want. Specific topics covered include rings arising from matrices, differential operators, and Lie algebras. Contains extensive references.

Noncommutative Noetherian rings

The Jacobson Radical of Group Algebras

The Jacobson Radical of Group Algebras

Let G be a finite group and let F be a field. It is well known that linear representations of G over F can be interpreted as modules over the group algebra FG. Thus the investigation of ring-theoretic structure of the Jacobson radical J(FG) of FG is of fundamental importance. During the last two decades the subject has been pursued by a number of researchers and many interesting results have been obtained. This volume examines these results.

The Jacobson Radical of Group Algebras