Commutative Rings

Author : Fanggui Wang,Hwankoo Kim
Publisher : Springer
Page : 699 pages
File Size : 54,6 Mb
Release : 2017-01-06
Category : Mathematics
ISBN : 9789811033377

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Foundations of Commutative Rings and Their Modules by Fanggui Wang,Hwankoo Kim Pdf

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.

Author : Frank De Meyer,Edward Ingraham
Publisher : Springer
Page : 162 pages
File Size : 50,8 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540364849

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Separable Algebras over Commutative Rings by Frank De Meyer,Edward Ingraham Pdf

These lecture notes were prepared by the authors for use in graduate courses and seminars, based on the work of many earlier mathematicians. In addition to very elementary results, presented for the convenience of the reader, Chapter I contains the Morita theorems and the definition of the projective class group of a commutative ring. Chapter II addresses the Brauer group of a commutative ring, and automorphisms of separable algebras. Chapter III surveys the principal theorems of the Galois theory for commutative rings. In Chapter IV the authors present a direct derivation of the first six terms of the seven-term exact sequence for Galois cohomology. In the fifth and final chapter the authors illustrate the preceding material with applications to the structure of central simple algebras and the Brauer group of a Dedekind domain, and they pose problems for further investigation. Exercises are included at the end of each chapter.

Author : Bernard R. McDonald
Publisher : CRC Press
Page : 563 pages
File Size : 44,9 Mb
Release : 2020-11-26
Category : Mathematics
ISBN : 9781000146462

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Linear Algebra over Commutative Rings by Bernard R. McDonald Pdf

This monograph arose from lectures at the University of Oklahoma on topics related to linear algebra over commutative rings. It provides an introduction of matrix theory over commutative rings. The monograph discusses the structure theory of a projective module.

Author : Robert Gilmer
Publisher : University of Chicago Press
Page : 392 pages
File Size : 51,9 Mb
Release : 1984-03-15
Category : Mathematics
ISBN : 9780226293929

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Commutative Semigroup Rings by Robert Gilmer Pdf

Commutative Semigroup Rings was the first exposition of the basic properties of semigroup rings. Gilmer concentrates on the interplay between semigroups and rings, thereby illuminating both of these important concepts in modern algebra.

Author : Hideyuki Matsumura
Publisher : Cambridge University Press
Page : 338 pages
File Size : 45,9 Mb
Release : 1989-05-25
Category : Mathematics
ISBN : 0521367646

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Commutative Ring Theory by Hideyuki Matsumura Pdf

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

Author : Ayman Badawi
Publisher : Nova Publishers
Page : 220 pages
File Size : 53,5 Mb
Release : 2006
Category : Mathematics
ISBN : 1600210651

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Focus on Commutative Rings Research by Ayman Badawi Pdf

Focus on Commutative Rings Research

Author : Harry C. Hutchins
Publisher : Unknown
Page : 184 pages
File Size : 41,9 Mb
Release : 1981
Category : Mathematics
ISBN : UCAL:B4407358

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Examples of Commutative Rings by Harry C. Hutchins Pdf

Author : Steven T. Dougherty
Publisher : Springer
Page : 103 pages
File Size : 40,6 Mb
Release : 2017-07-04
Category : Mathematics
ISBN : 9783319598062

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Algebraic Coding Theory Over Finite Commutative Rings by Steven T. Dougherty Pdf

This book provides a self-contained introduction to algebraic coding theory over finite Frobenius rings. It is the first to offer a comprehensive account on the subject. Coding theory has its origins in the engineering problem of effective electronic communication where the alphabet is generally the binary field. Since its inception, it has grown as a branch of mathematics, and has since been expanded to consider any finite field, and later also Frobenius rings, as its alphabet. This book presents a broad view of the subject as a branch of pure mathematics and relates major results to other fields, including combinatorics, number theory and ring theory. Suitable for graduate students, the book will be of interest to anyone working in the field of coding theory, as well as algebraists and number theorists looking to apply coding theory to their own work.

Author : Irving Kaplansky
Publisher : Unknown
Page : 200 pages
File Size : 51,5 Mb
Release : 1970
Category : Commutative rings
ISBN : UOM:49015001038687

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Commutative Rings by Irving Kaplansky Pdf

Author : David F. Anderson,David Dobbs
Publisher : CRC Press
Page : 820 pages
File Size : 41,5 Mb
Release : 1995-04-10
Category : Mathematics
ISBN : 0824796055

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Zero-Dimensional Commutative Rings by David F. Anderson,David Dobbs Pdf

This work presents advances in zero-dimensional commutative rings and commutative algebra. It illustrates the research frontier with 52 open problems together with comments on the relevant literature, and offers a comprehensive index for easy access to information. Wide-ranging developments in commutative ring theory are examined.

Author : T.Y. Lam
Publisher : Springer Science & Business Media
Page : 410 pages
File Size : 52,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781468404067

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A First Course in Noncommutative Rings by T.Y. Lam Pdf

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.

Author : Gilberto Bini,Flaminio Flamini
Publisher : Springer Science & Business Media
Page : 181 pages
File Size : 45,5 Mb
Release : 2012-12-06
Category : Technology & Engineering
ISBN : 9781461509578

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Finite Commutative Rings and Their Applications by Gilberto Bini,Flaminio Flamini Pdf

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.

Author : K.P.S. Bhaskara Rao
Publisher : CRC Press
Page : 192 pages
File Size : 45,8 Mb
Release : 2002-03-21
Category : Mathematics
ISBN : 9780203218877

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Theory of Generalized Inverses Over Commutative Rings by K.P.S. Bhaskara Rao Pdf

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

Author : Ihsen Yengui
Publisher : Springer
Page : 277 pages
File Size : 49,6 Mb
Release : 2015-12-11
Category : Mathematics
ISBN : 9783319194943

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Constructive Commutative Algebra by Ihsen Yengui Pdf

The main goal of this book is to find the constructive content hidden in abstract proofs of concrete theorems in Commutative Algebra, especially in well-known theorems concerning projective modules over polynomial rings (mainly the Quillen-Suslin theorem) and syzygies of multivariate polynomials with coefficients in a valuation ring. Simple and constructive proofs of some results in the theory of projective modules over polynomial rings are also given, and light is cast upon recent progress on the Hermite ring and Gröbner ring conjectures. New conjectures on unimodular completion arising from our constructive approach to the unimodular completion problem are presented. Constructive algebra can be understood as a first preprocessing step for computer algebra that leads to the discovery of general algorithms, even if they are sometimes not efficient. From a logical point of view, the dynamical evaluation gives a constructive substitute for two highly nonconstructive tools of abstract algebra: the Law of Excluded Middle and Zorn's Lemma. For instance, these tools are required in order to construct the complete prime factorization of an ideal in a Dedekind ring, whereas the dynamical method reveals the computational content of this construction. These lecture notes follow this dynamical philosophy.

Author : S.T. Chapman,Sarah Glaz
Publisher : Springer Science & Business Media
Page : 477 pages
File Size : 43,7 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9781475731804

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Non-Noetherian Commutative Ring Theory by S.T. Chapman,Sarah Glaz Pdf

Commutative Ring Theory emerged as a distinct field of research in math ematics only at the beginning of the twentieth century. It is rooted in nine teenth century major works in Number Theory and Algebraic Geometry for which it provided a useful tool for proving results. From this humble origin, it flourished into a field of study in its own right of an astonishing richness and interest. Nowadays, one has to specialize in an area of this vast field in order to be able to master its wealth of results and come up with worthwhile contributions. One of the major areas of the field of Commutative Ring Theory is the study of non-Noetherian rings. The last ten years have seen a lively flurry of activity in this area, including: a large number of conferences and special sections at national and international meetings dedicated to presenting its results, an abundance of articles in scientific journals, and a substantial number of books capturing some of its topics. This rapid growth, and the occasion of the new Millennium, prompted us to embark on a project aimed at presenting an overview of the recent research in the area. With this in mind, we invited many of the most prominent researchers in Non-Noetherian Commutative Ring Theory to write expository articles representing the most recent topics of research in this area.