Finite Commutative Rings and Their Applications

Finite Commutative Rings and Their Applications
Author: Gilberto Bini
Publisher: Springer Science & Business Media
Total Pages: 181
Release: 2012-12-06
Genre: Technology & Engineering
ISBN: 1461509572

Foreword by Dieter Jungnickel Finite Commutative Rings and their Applications answers a need for an introductory reference in finite commutative ring theory as applied to information and communication theory. This book will be of interest to both professional and academic researchers in the fields of communication and coding theory. The book is a concrete and self-contained introduction to finite commutative local rings, focusing in particular on Galois and Quasi-Galois rings. The reader is provided with an active and concrete approach to the study of the purely algebraic structure and properties of finite commutative rings (in particular, Galois rings) as well as to their applications to coding theory. Finite Commutative Rings and their Applications is the first to address both theoretical and practical aspects of finite ring theory. The authors provide a practical approach to finite rings through explanatory examples, thereby avoiding an abstract presentation of the subject. The section on Quasi-Galois rings presents new and unpublished results as well. The authors then introduce some applications of finite rings, in particular Galois rings, to coding theory, using a solid algebraic and geometric theoretical background.

Commutative Ring Theory

Commutative Ring Theory
Author: Hideyuki Matsumura
Publisher: Cambridge University Press
Total Pages: 338
Release: 1989-05-25
Genre: Mathematics
ISBN: 9780521367646

This book explores commutative ring theory, an important a foundation for algebraic geometry and complex analytical geometry.

Commutative Ring Theory and Applications

Commutative Ring Theory and Applications
Author: Marco Fontana
Publisher: CRC Press
Total Pages: 524
Release: 2017-07-27
Genre: Mathematics
ISBN: 9780203910627

Featuring presentations from the Fourth International Conference on Commutative Algebra held in Fez, Morocco, this reference presents trends in the growing area of commutative algebra. With contributions from nearly 50 internationally renowned researchers, the book emphasizes innovative applications and connections to algebraic number theory, geome

Theory of Generalized Inverses Over Commutative Rings

Theory of Generalized Inverses Over Commutative Rings
Author: K.P.S. Bhaskara Rao
Publisher: CRC Press
Total Pages: 192
Release: 2002-03-21
Genre: Mathematics
ISBN: 0203218876

The theory of generalized inverses of real or complex matrices has been expertly developed and documented. But the generalized inverses of matrices over rings have received comprehensive treatment only recently. In this book, the author, who contributed to the research and development of the theory, explains his results. He explores regular element

Non-Noetherian Commutative Ring Theory

Non-Noetherian Commutative Ring Theory
Author: S.T. Chapman
Publisher: Springer Science & Business Media
Total Pages: 504
Release: 2000-10-31
Genre: Mathematics
ISBN: 9780792364924

This volume consists of twenty-one articles by many of the most prominent researchers in non-Noetherian commutative ring theory. The articles combine in various degrees surveys of past results, recent results that have never before seen print, open problems, and an extensive bibliography. One hundred open problems supplied by the authors have been collected in the volume's concluding chapter. The entire collection provides a comprehensive survey of the development of the field over the last ten years and points to future directions of research in the area. Audience: Researchers and graduate students; the volume is an appropriate source of material for several semester-long graduate-level seminars and courses.

Foundations of Commutative Rings and Their Modules

Foundations of Commutative Rings and Their Modules
Author: Fanggui Wang
Publisher: Springer
Total Pages: 714
Release: 2017-01-06
Genre: Mathematics
ISBN: 9811033374

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.

Algorithmic Methods in Non-Commutative Algebra

Algorithmic Methods in Non-Commutative Algebra
Author: J.L. Bueso
Publisher: Springer Science & Business Media
Total Pages: 307
Release: 2013-03-09
Genre: Computers
ISBN: 9401702853

The already broad range of applications of ring theory has been enhanced in the eighties by the increasing interest in algebraic structures of considerable complexity, the so-called class of quantum groups. One of the fundamental properties of quantum groups is that they are modelled by associative coordinate rings possessing a canonical basis, which allows for the use of algorithmic structures based on Groebner bases to study them. This book develops these methods in a self-contained way, concentrating on an in-depth study of the notion of a vast class of non-commutative rings (encompassing most quantum groups), the so-called Poincaré-Birkhoff-Witt rings. We include algorithms which treat essential aspects like ideals and (bi)modules, the calculation of homological dimension and of the Gelfand-Kirillov dimension, the Hilbert-Samuel polynomial, primality tests for prime ideals, etc.

Introduction To Commutative Algebra

Introduction To Commutative Algebra
Author: Michael F. Atiyah
Publisher: CRC Press
Total Pages: 140
Release: 2018-03-09
Genre: Mathematics
ISBN: 0429973268

First Published in 2018. This book grew out of a course of lectures given to third year undergraduates at Oxford University and it has the modest aim of producing a rapid introduction to the subject. It is designed to be read by students who have had a first elementary course in general algebra. On the other hand, it is not intended as a substitute for the more voluminous tracts such as Zariski-Samuel or Bourbaki. We have concentrated on certain central topics, and large areas, such as field theory, are not touched. In content we cover rather more ground than Northcott and our treatment is substantially different in that, following the modern trend, we put more emphasis on modules and localization.

Steps in Commutative Algebra

Steps in Commutative Algebra
Author: R. Y. Sharp
Publisher: Cambridge University Press
Total Pages: 371
Release: 2000
Genre: Mathematics
ISBN: 0521646235

Introductory account of commutative algebra, aimed at students with a background in basic algebra.