Algebras, Rings and Modules

Author: Michiel Hazewinkel,Nadiâ Mihajlivna Gubareni,Vladimir Vasil'evič Kiričenko

Publisher: Springer Science & Business Media

ISBN: 1402051409

Category: Modules (Algebra)

Page: 400

View: 7575

Algebras, Rings and Modules, Volume 2

Non-commutative Algebras and Rings

Author: Michiel Hazewinkel,Nadiya M. Gubareni

Publisher: CRC Press

ISBN: 1351869876

Category: Mathematics

Page: 364

View: 1395

The theory of algebras, rings, and modules is one of the fundamental domains of modern mathematics. General algebra, more specifically non-commutative algebra, is poised for major advances in the twenty-first century (together with and in interaction with combinatorics), just as topology, analysis, and probability experienced in the twentieth century. This is the second volume of Algebras, Rings and Modules: Non-commutative Algebras and Rings by M. Hazewinkel and N. Gubarenis, a continuation stressing the more important recent results on advanced topics of the structural theory of associative algebras, rings and modules.

Topics in Hyperplane Arrangements

Author: Marcelo Aguiar,Swapneel Mahajan

Publisher: American Mathematical Soc.

ISBN: 1470437112

Category: Algebraic spaces

Page: 611

View: 1857

This monograph studies the interplay between various algebraic, geometric and combinatorial aspects of real hyperplane arrangements. It provides a careful, organized and unified treatment of several recent developments in the field, and brings forth many new ideas and results. It has two parts, each divided into eight chapters, and five appendices with background material. Part I gives a detailed discussion on faces, flats, chambers, cones, gallery intervals, lunes and other geometric notions associated with arrangements. The Tits monoid plays a central role. Another important object is the category of lunes which generalizes the classical associative operad. Also discussed are the descent and lune identities, distance functions on chambers, and the combinatorics of the braid arrangement and related examples. Part II studies the structure and representation theory of the Tits algebra of an arrangement. It gives a detailed analysis of idempotents and Peirce decompositions, and connects them to the classical theory of Eulerian idempotents. It introduces the space of Lie elements of an arrangement which generalizes the classical Lie operad. This space is the last nonzero power of the radical of the Tits algebra. It is also the socle of the left ideal of chambers and of the right ideal of Zie elements. Zie elements generalize the classical Lie idempotents. They include Dynkin elements associated to generic half-spaces which generalize the classical Dynkin idempotent. Another important object is the lune-incidence algebra which marks the beginning of noncommutative Möbius theory. These ideas are also brought upon the study of the Solomon descent algebra. The monograph is written with clarity and in sufficient detail to make it accessible to graduate students. It can also serve as a useful reference to experts.

Abstract Algebra with Applications

Volume 2: Rings and Fields

Author: Karlheinz Spindler

Publisher: CRC Press

ISBN: 9780824791599

Category: Mathematics

Page: 552

View: 8106

A comprehensive presentation of abstract algebra and an in-depth treatment of the applications of algebraic techniques and the relationship of algebra to other disciplines, such as number theory, combinatorics, geometry, topology, differential equations, and Markov chains.

Semidistributive Modules and Rings

Author: A.A. Tuganbaev

Publisher: Springer Science & Business Media

ISBN: 9401150869

Category: Mathematics

Page: 357

View: 2226

A module M is called distributive if the lattice Lat(M) of all its submodules is distributive, i.e., Fn(G + H) = FnG + FnH for all submodules F,G, and H of the module M. A module M is called uniserial if all its submodules are comparable with respect to inclusion, i.e., the lattice Lat(M) is a chain. Any direct sum of distributive (resp. uniserial) modules is called a semidistributive (resp. serial) module. The class of distributive (resp. semidistributive) modules properly cont.ains the class ofall uniserial (resp. serial) modules. In particular, all simple (resp. semisimple) modules are distributive (resp. semidistributive). All strongly regular rings (for example, all factor rings of direct products of division rings and all commutative regular rings) are distributive; all valuation rings in division rings and all commutative Dedekind rings (e.g., rings of integral algebraic numbers or commutative principal ideal rings) are distributive. A module is called a Bezout module or a locally cyclic module ifevery finitely generated submodule is cyclic. If all maximal right ideals of a ring A are ideals (e.g., if A is commutative), then all Bezout A-modules are distributive.

Approximations and Endomorphism Algebras of Modules

Volume 1 – Approximations / Volume 2 – Predictions

Author: Rüdiger Göbel,Jan Trlifaj

Publisher: Walter de Gruyter

ISBN: 3110218119

Category: Mathematics

Page: 1024

View: 7997

This monograph – now in its second revised and extended edition – provides a thorough treatment of module theory, a subfield of algebra. The authors develop an approximation theory as well as realization theorems and present some of its recent applications, notably to infinite-dimensional combinatorics and model theory. The book starts from basic facts and gradually develops the theory towards its present frontiers.

Kategorien und Funktoren

Author: Bodo Pareigis

Publisher: N.A


Category: Categories (Mathematics)

Page: 192

View: 3697

Basic Algebra

Groups, Rings and Fields

Author: P.M. Cohn

Publisher: Springer Science & Business Media

ISBN: 0857294288

Category: Mathematics

Page: 465

View: 2565

This is the first volume of a revised edition of P.M. Cohn's classic three-volume text Algebra, widely regarded as one of the most outstanding introductory algebra textbooks. This volume covers the important results of algebra. Readers should have some knowledge of linear algebra, groups and fields, although all the essential facts and definitions are recalled.

Extensions of Rings and Modules

Author: Gary F. Birkenmeier,Jae Keol Park,S Tariq Rizvi

Publisher: Springer Science & Business Media

ISBN: 0387927166

Category: Mathematics

Page: 432

View: 9228

The "extensions" of rings and modules have yet to be explored in detail in a research monograph. This book presents state of the art research and also stimulating new and further research. Broken into three parts, Part I begins with basic notions, terminology, definitions and a description of the classes of rings and modules. Part II considers the transference of conditions between a base ring or module and its extensions. And Part III utilizes the concept of a minimal essental extension with respect to a specific class (a hull). Mathematical interdisciplinary applications appear throughout. Major applications of the ring and module theory to Functional Analysis, especially C*-algebras, appear in Part III, make this book of interest to Algebra and Functional Analysis researchers. Notes and exercises at the end of every chapter, and open problems at the end of all three parts, lend this as an ideal textbook for graduate or advanced undergradate students.

Commutative Algebra II

Author: O. Zariski,P. Samuel

Publisher: Springer Science & Business Media

ISBN: 9780387901718

Category: Mathematics

Page: 414

View: 1820

From the Preface: "topics are: (a) valuation theory; (b) theory of polynomial and power series rings (including generalizations to graded rings and modules); (c) local algebra... the algebro-geometric connections and applications of the purely algebraic material are constantly stressed and abundantly scattered throughout the exposition. Thus, this volume can be used in part as an introduction to some basic concepts and the arithmetic foundations of algebraic geometry."

Algebra II

Noncommutative Rings Identities

Author: A.I. Kostrikin,I.R. Shafarevich

Publisher: Springer Science & Business Media

ISBN: 3642728995

Category: Mathematics

Page: 234

View: 6979

The algebra of square matrices of size n ~ 2 over the field of complex numbers is, evidently, the best-known example of a non-commutative alge 1 bra • Subalgebras and subrings of this algebra (for example, the ring of n x n matrices with integral entries) arise naturally in many areas of mathemat ics. Historically however, the study of matrix algebras was preceded by the discovery of quatemions which, introduced in 1843 by Hamilton, found ap plications in the classical mechanics of the past century. Later it turned out that quaternion analysis had important applications in field theory. The al gebra of quaternions has become one of the classical mathematical objects; it is used, for instance, in algebra, geometry and topology. We will briefly focus on other examples of non-commutative rings and algebras which arise naturally in mathematics and in mathematical physics. The exterior algebra (or Grassmann algebra) is widely used in differential geometry - for example, in geometric theory of integration. Clifford algebras, which include exterior algebras as a special case, have applications in rep resentation theory and in algebraic topology. The Weyl algebra (Le. algebra of differential operators with· polynomial coefficients) often appears in the representation theory of Lie algebras. In recent years modules over the Weyl algebra and sheaves of such modules became the foundation of the so-called microlocal analysis. The theory of operator algebras (Le.

Introductory Lectures on Rings and Modules

Author: John A. Beachy

Publisher: Cambridge University Press

ISBN: 9780521644075

Category: Mathematics

Page: 238

View: 4126

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

commutative ring theory

Proceedings of the Ii International Conference

Author: Paul-Jean Cahen,Marco Fontana,Evan Houston,Salah-Eddine Kabbaj

Publisher: CRC Press

ISBN: 9780824798154

Category: Mathematics

Page: 488

View: 9383

Presents the proceedings of the Second International Conference on Commutative Ring Theory in Fes, Morocco. The text details developments in commutative algebra, highlighting the theory of rings and ideals. It explores commutative algebra's connections with and applications to topological algebra and algebraic geometry.

Algebraic K-Theory and Its Applications

Author: Jonathan Rosenberg

Publisher: Springer Science & Business Media

ISBN: 9780387942483

Category: Mathematics

Page: 392

View: 9902

Algebraic K-Theory plays an important role in many areas of modern mathematics: most notably algebraic topology, number theory, and algebraic geometry, but even including operator theory. The broad range of these topics has tended to give the subject an aura of inapproachability. This book, based on a course at the University of Maryland in the fall of 1990, is intended to enable graduate students or mathematicians working in other areas not only to learn the basics of algebraic K-Theory, but also to get a feel for its many applications. The required prerequisites are only the standard one-year graduate algebra course and the standard introductory graduate course on algebraic and geometric topology. Many topics from algebraic topology, homological algebra, and algebraic number theory are developed as needed. The final chapter gives a concise introduction to cyclic homology and its interrelationship with K-Theory.

Model Theoretic Algebra With Particular Emphasis on Fields, Rings, Modules

Author: Christian. U Jensen,Helmt Lenzing

Publisher: CRC Press

ISBN: 9782881247170

Category: Mathematics

Page: 464

View: 1337

Looks like a text (and a handsome one at that), but the authors prefer to describe their creation as "notes", intended to acquaint graduate students with "the power of the most basic principles of model theory by applying them to classical questions in algebra". Thirteen chapters (the last given to the enumeration of some open problems), plus tables and several appendices, bibliography. (NW) Annotation copyrighted by Book News, Inc., Portland, OR

Integration and Probability

Author: Paul Malliavin

Publisher: Springer Science & Business Media

ISBN: 1461242029

Category: Mathematics

Page: 326

View: 3416

An introduction to analysis with the right mix of abstract theories and concrete problems. Starting with general measure theory, the book goes on to treat Borel and Radon measures and introduces the reader to Fourier analysis in Euclidean spaces with a treatment of Sobolev spaces, distributions, and the corresponding Fourier analysis. It continues with a Hilbertian treatment of the basic laws of probability including Doob's martingale convergence theorem and finishes with Malliavin's "stochastic calculus of variations" developed in the context of Gaussian measure spaces. This invaluable contribution gives a taste of the fact that analysis is not a collection of independent theories, but can be treated as a whole.

Problems and Theorems in Analysis II

Theory of Functions. Zeros. Polynomials. Determinants. Number Theory. Geometry

Author: George Polya,Gabor Szegö

Publisher: Springer Science & Business Media

ISBN: 9783540636861

Category: Mathematics

Page: 392

View: 5761

Few mathematical books are worth translating 50 years after original publication. Polyá-Szegö is one! It was published in German in 1924, and its English edition was widely acclaimed when it appeared in 1972. In the past, more of the leading mathematicians proposed and solved problems than today. Their collection of the best in analysis is a heritage of lasting value.

A Guide to the Literature on Semirings and their Applications in Mathematics and Information Sciences

With Complete Bibliography

Author: K. Glazek

Publisher: Springer Science & Business Media

ISBN: 9781402007170

Category: Mathematics

Page: 392

View: 9448

This book presents a guide to the extensive literature on the topic of semirings and includes a complete bibliography. It serves as a complement to the existing monographs and a point of reference to researchers and students on this topic. The literature on semirings has evolved over many years, in a variety of languages, by authors representing different schools of mathematics and working in various related fields. Recently, semiring theory has experienced rapid development, although publications are widely scattered. This survey also covers those newly emerged areas of semiring applications that have not received sufficient treatment in widely accessible monographs, as well as many lesser-known or `forgotten' works. The author has been collecting the bibliographic data for this book since 1985. Over the years, it has proved very useful for specialists. For example, J.S. Golan wrote he owed `... a special debt to Kazimierz Glazek, whose bibliography proved to be an invaluable guide to the bewildering maze of literature on semirings'. U. Hebisch and H.J. Weinert also mentioned his collection of literature had been of great assistance to them. Now updated to include publications up to the beginning of 2002, this work is available to a wide readership. Audience: This volume is the first single reference that can guide the interested scholar or student to the relevant publications in semirings, semifields, algebraic theory of languages and automata, positive matrices and other generalisations, and ordered semigroups and groups.