Algebras Rings And Modules
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Author | : Michiel Hazewinkel |
Publisher | : Springer Science & Business Media |
Total Pages | : 393 |
Release | : 2006-01-18 |
Genre | : Mathematics |
ISBN | : 1402026919 |
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?rstexampleofanoncommutativeā€¯numbersystemā€¯. 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.
Author | : Frank W. Anderson |
Publisher | : Springer Science & Business Media |
Total Pages | : 386 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 1461244188 |
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 | : Michiel Hazewinkel |
Publisher | : CRC Press |
Total Pages | : 384 |
Release | : 2016-04-05 |
Genre | : Mathematics |
ISBN | : 1482245051 |
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 centu
Author | : T.Y. Lam |
Publisher | : Springer Science & Business Media |
Total Pages | : 427 |
Release | : 2009-12-08 |
Genre | : Mathematics |
ISBN | : 0387488995 |
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 | : Alberto Facchini |
Publisher | : CRC Press |
Total Pages | : 530 |
Release | : 2020-02-10 |
Genre | : Mathematics |
ISBN | : 9780824750817 |
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 | : Craig Huneke |
Publisher | : Cambridge University Press |
Total Pages | : 446 |
Release | : 2006-10-12 |
Genre | : Mathematics |
ISBN | : 0521688604 |
Ideal for graduate students and researchers, this book presents a unified treatment of the central notions of integral closure.
Author | : Tsit-Yuen Lam |
Publisher | : Springer Science & Business Media |
Total Pages | : 577 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 1461205255 |
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 | : John Dauns |
Publisher | : Cambridge University Press |
Total Pages | : 470 |
Release | : 1994-10-28 |
Genre | : Mathematics |
ISBN | : 0521462584 |
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 | : Joachim Lambek |
Publisher | : |
Total Pages | : 206 |
Release | : 1966 |
Genre | : Associative rings |
ISBN | : |
Author | : Toma Albu |
Publisher | : Springer Science & Business Media |
Total Pages | : 204 |
Release | : 2011-02-04 |
Genre | : Mathematics |
ISBN | : 3034600070 |
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.