Classical Artinian Rings And Related Topics

Author : Yoshitomo Baba,Kiyoichi ?shiro
Publisher : World Scientific
Page : 310 pages
File Size : 53,6 Mb
Release : 2009
Category : Mathematics
ISBN : 9789814287241

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Classical Artinian Rings and Related Topics by Yoshitomo Baba,Kiyoichi ?shiro Pdf

Quasi-Frobenius rings and Nakayama rings were introduced by T Nakayama in 1939. Since then, these classical artinian rings have continued to fascinate ring theorists with their abundance of properties and structural depth. In 1978, M Harada introduced a new class of artinian rings which were later called Harada rings in his honour. Quasi-Frobenius rings, Nakayama rings and Harada rings are very closely interrelated. As a result, from a new perspective, we may study the classical artinian rings through their interaction and overlap with Harada rings. The objective of this seminal work is to present the structure of Harada rings and provide important applications of this structure to the classical artinian rings. In the process, we cover many topics on artinian rings, using a wide variety of concepts from the theory of rings and modules. In particular, we consider the following topics, all of which are currently of much interest and ongoing research: Nakayama permutations, Nakayama automorphisms, Fuller's theorem on i-pairs, artinian rings with self-duality, skew-matrix rings, the classification of Nakayama rings, Nakayama group algebras, the Faith conjecture, constructions of local quasi-Frobenius rings, lifting modules, and extending modules. In our presentation of these topics, the reader will be able to retrace the history of artinian rings.

Author : Yoshitomo Baba
Publisher : World Scientific
Page : 310 pages
File Size : 43,5 Mb
Release : 2009
Category : Mathematics
ISBN : 9789814287258

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Classical Artinian Rings and Related Topics by Yoshitomo Baba Pdf

Quasi-Frobenius rings and Nakayama rings were introduced by T Nakayama in 1939. Since then, these classical artinian rings have continued to fascinate ring theorists with their abundance of properties and structural depth. In 1978, M Harada introduced a new class of artinian rings which were later called Harada rings in his honour. Quasi-Frobenius rings, Nakayama rings and Harada rings are very closely interrelated. As a result, from a new perspective, we may study the classical artinian rings through their interaction and overlap with Harada rings. The objective of this seminal work is to present the structure of Harada rings and provide important applications of this structure to the classical artinian rings. In the process, we cover many topics on artinian rings, using a wide variety of concepts from the theory of rings and modules. In particular, we consider the following topics, all of which are currently of much interest and ongoing research : Nakayama permutations, Nakayama automorphisms, Fuller's theorem on i-pairs, artinian rings with self-duality, skew-matrix rings, the classification of Nakayama rings, Nakayama group algebras, the Faith conjecture, constructions of local quasi-Frobenius rings, lifting modules, and extending modules. In our presentation of these topics, the reader will be able to retrace the history of artinian rings.

Author : Hidetoshi Marubayashi
Publisher : World Scientific
Page : 314 pages
File Size : 44,9 Mb
Release : 2009
Category : Mathematics
ISBN : 9789812818331

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Ring Theory 2007 by Hidetoshi Marubayashi Pdf

This volume consists of a collection of survey articles by invited speakers and original articles refereed by world experts that was presented at the fifth ChinaOCoJapanOCoKorea International Symposium. The survey articles provide some ideas of the application as well as an excellent overview of the various areas in ring theory. The original articles exhibit new ideas, tools and techniques needed for successful research investigation in ring theory and show the trend of current research."

Author : Hideto Asashiba,Shigeto Kawata,Nanqing Ding,Nam Kyun Kim
Publisher : World Scientific
Page : 256 pages
File Size : 41,6 Mb
Release : 2021-01-04
Category : Mathematics
ISBN : 9789811230301

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Ring Theory 2019 - Proceedings Of The Eighth China-japan-korea International Symposium On Ring Theory by Hideto Asashiba,Shigeto Kawata,Nanqing Ding,Nam Kyun Kim Pdf

Since 1991, the group of ring theorists from China and Japan, joined by Korea from 1995 onwards, took turns to hold the quadrennial international conferences (sometimes also referred to as symposiums). As the proceedings of the eighth conference held in Nagoya, Japan in 2019, this volume consists of a collection of articles by invited speakers (survey) and general speakers (survey and original), all of which were refereed by world experts.The survey articles show the trends of current research and offer clear, thorough explanations that are ideal for researchers also in other specialized areas of ring theory. The original articles display new results, ideas and tools for research investigations in ring theory.The articles cover major areas in ring theory, such as: structures of rings, module theory, homological algebra, groups, Hopf algebras, Lie theory, representation theory of rings, (non-commutative) algebraic geometry, commutative rings (structures, representations), amongst others.This volume is a useful resource for researchers — both beginners and advanced experts — in ring theory.

Author : Dinh Van Huynh, S. K. Jain, Sergio R. López-Permouth,S. Tariq Rizvi,Cosmin S. Roman
Publisher : American Mathematical Soc.
Page : 330 pages
File Size : 54,5 Mb
Release : 2014-02-21
Category : Mathematics
ISBN : 9780821887974

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Ring Theory and Its Applications by Dinh Van Huynh, S. K. Jain, Sergio R. López-Permouth,S. Tariq Rizvi,Cosmin S. Roman Pdf

This volume contains the proceedings of the Ring Theory Session in honor of T. Y. Lam's 70th birthday, at the 31st Ohio State-Denison Mathematics Conference, held from May 25-27, 2012, at The Ohio State University, Columbus, Ohio. Included are expository articles and research papers covering topics such as cyclically presented modules, Eggert's conjecture, the Mittag-Leffler conditions, clean rings, McCoy rings, QF rings, projective and injective modules, Baer modules, and Leavitt path algebras. Graduate students and researchers in many areas of algebra will find this volume valuable as the papers point out many directions for future work; in particular, several articles contain explicit lists of open questions.

Author : Piotr Krylov,Askar Tuganbaev
Publisher : Springer
Page : 156 pages
File Size : 52,6 Mb
Release : 2017-03-30
Category : Mathematics
ISBN : 9783319539072

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Formal Matrices by Piotr Krylov,Askar Tuganbaev Pdf

This monograph is a comprehensive account of formal matrices, examining homological properties of modules over formal matrix rings and summarising the interplay between Morita contexts and K theory. While various special types of formal matrix rings have been studied for a long time from several points of view and appear in various textbooks, for instance to examine equivalences of module categories and to illustrate rings with one-sided non-symmetric properties, this particular class of rings has, so far, not been treated systematically. Exploring formal matrix rings of order 2 and introducing the notion of the determinant of a formal matrix over a commutative ring, this monograph further covers the Grothendieck and Whitehead groups of rings. Graduate students and researchers interested in ring theory, module theory and operator algebras will find this book particularly valuable. Containing numerous examples, Formal Matrices is a largely self-contained and accessible introduction to the topic, assuming a solid understanding of basic algebra.

Author : T.Y. Lam
Publisher : Springer Science & Business Media
Page : 299 pages
File Size : 45,5 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9781475739879

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Exercises in Classical Ring Theory by T.Y. Lam Pdf

Based in large part on the comprehensive "First Course in Ring Theory" by the same author, this book provides a comprehensive set of problems and solutions in ring theory that will serve not only as a teaching aid to instructors using that book, but also for students, who will see how ring theory theorems are applied to solving ring-theoretic problems and how good proofs are written. The author demonstrates that problem-solving is a lively process: in "Comments" following many solutions he discusses what happens if a hypothesis is removed, whether the exercise can be further generalized, what would be a concrete example for the exercise, and so forth. The book is thus much more than a solution manual.

Author : Anonim
Publisher : Unknown
Page : 486 pages
File Size : 41,5 Mb
Release : 2015
Category : Mathematics
ISBN : IND:30000153092204

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Hokkaido Mathematical Journal by Anonim Pdf

Author : Frank W. Anderson,Kent R. Fuller
Publisher : Springer Science & Business Media
Page : 386 pages
File Size : 51,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461244189

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Rings and Categories of Modules by Frank W. Anderson,Kent R. Fuller Pdf

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 : T.Y. Lam
Publisher : Springer Science & Business Media
Page : 427 pages
File Size : 50,6 Mb
Release : 2009-12-08
Category : Mathematics
ISBN : 9780387488998

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Exercises in Modules and Rings by T.Y. Lam Pdf

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 : Martin Mathieu
Publisher : Springer Nature
Page : 159 pages
File Size : 48,5 Mb
Release : 2022-12-19
Category : Mathematics
ISBN : 9783031142093

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Classically Semisimple Rings by Martin Mathieu Pdf

Classically Semisimple Rings is a textbook on rings, modules and categories, aimed at advanced undergraduate and beginning graduate students. The book presents the classical theory of semisimple rings from a modern, category-theoretic point of view. Examples from algebra are used to motivate the abstract language of category theory, which then provides a framework for the study of rings and modules, culminating in the Wedderburn–Artin classification of semisimple rings. In the last part of the book, readers are gently introduced to related topics such as tensor products, exchange modules and C*-algebras. As a final flourish, Rickart’s theorem on group rings ties a number of these topics together. Each chapter ends with a selection of exercises of varying difficulty, and readers interested in the history of mathematics will find biographical sketches of important figures scattered throughout the text. Assuming previous knowledge in linear and basic abstract algebra, this book can serve as a textbook for a course in algebra, providing students with valuable early exposure to category theory.

Author : Gary F. Birkenmeier,Jae K. Park,Young S. Park
Publisher : Springer Science & Business Media
Page : 449 pages
File Size : 49,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461201816

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International Symposium on Ring Theory by Gary F. Birkenmeier,Jae K. Park,Young S. Park Pdf

This volume is the Proceedings of the Third Korea-China-Japan Inter national Symposium on Ring Theory held jointly with the Second Korea Japan Joint Ring Theory Seminar which took place at the historical resort area of Korea, Kyongju, June 28-July 3, 1999. It also includes articles by some invited mathematicians who were unable to attend the conference. Over 90 mathematicians from 12 countries attended this conference. The conference is held every 4 years on a rotating basis. The first con ference was held in 1991 at Guilin, China. In 1995 the second conference took place in Okayama, Japan. At the second conference it was decided to include Korea, who hosted this conference of 1999. During the past century Ring Theory has diversified into many subar eas. This is reflected in these articles from over 25 well-known mathemati cians covering a broad range of topics, including: Classical Ring Theory, Module Theory, Representation Theory, and the theory of Hopf Algebras. Among these peer reviewed papers are invited survey articles as well as research articles. The survey articles provide an overview of various areas for researchers looking for a new or related field to investigate, while the research articles give the flavor of current research. We feel that the variety of related topics will stimulate interaction between researchers. Moreover the Open Problems section provides guidance for future research. This book should prove attractive to a wide audience of algebraists. Gary F. Birkenmeier, Lafayette, U. S. A.

Author : Joachim Lambek
Publisher : American Mathematical Soc.
Page : 196 pages
File Size : 50,8 Mb
Release : 2009
Category : Associative rings
ISBN : 9780821849002

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Lectures on Rings and Modules by Joachim Lambek Pdf

This book is an introduction to the theory of associative rings and their modules, designed primarily for graduate students. The standard topics on the structure of rings are covered, with a particular emphasis on the concept of the complete ring of quotients. A survey of the fundamental concepts of algebras in the first chapter helps to make the treatment self-contained. The topics covered include selected results on Boolean and other commutative rings, the classical structure theory of associative rings, injective modules, and rings of quotients. The final chapter provides an introduction to homological algebra. Besides three appendices on further results, there is a six-page section of historical comments. Table of Contents: Fundamental Concepts of Algebra: 1.1 Rings and related algebraic systems; 1.2 Subrings, homomorphisms, ideals; 1.3 Modules, direct products, and direct sums; 1.4 Classical isomorphism theorems. Selected Topics on Commutative Rings: 2.1 Prime ideals in commutative rings; 2.2 Prime ideals in special commutative rings; 2.3 The complete ring of quotients of a commutative ring; 2.4 Rings of quotients of commutative semiprime rings; 2.5 Prime ideal spaces.Classical Theory of Associative Rings: 3.1 Primitive rings; 3.2 Radicals; 3.3 Completely reducible modules; 3.4 Completely reducible rings; 3.5 Artinian and Noetherian rings; 3.6 On lifting idempotents; 3.7 Local and semiperfect rings. Injectivity and Related Concepts: 4.1 Projective modules; 4.2 Injective modules; 4.3 The complete ring of quotients; 4.4 Rings of endomorphisms of injective modules; 4.5 Regular rings of quotients; 4.6 Classical rings of quotients; 4.7 The Faith-Utumi theorem. Introduction to Homological Algebra: 5.1 Tensor products of modules; 5.2 Hom and $\otimes$ as functors; 5.3 Exact sequences; 5.4 Flat modules; 5.5 Torsion and extension products. Appendixes; Comments; Bibliography; Index. Review from Zentralblatt Math: Due to their clarity and intelligible presentation, these lectures on rings and modules are a particularly successful introduction to the surrounding circle of ideas. Review from American Mathematical Monthly: An introduction to associative rings and modules which requires of the reader only the mathematical maturity which one would attain in a first-year graduate algebra [course]...in order to make the contents of the book as accessible as possible, the author develops all the fundamentals he will need.In addition to covering the basic topics...the author covers some topics not so readily available to the nonspecialist...the chapters are written to be as independent as possible...[which will be appreciated by] students making their first acquaintance with the subject...one of the most successful features of the book is that it can be read by graduate students with little or no help from a specialist. (CHEL/283.H)

Author : Vesselin Drensky,Antonio Giambruno,Sudarshan K. Sehgal
Publisher : CRC Press
Page : 332 pages
File Size : 42,7 Mb
Release : 1998-03-27
Category : Mathematics
ISBN : 0824701836

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Methods in Ring Theory by Vesselin Drensky,Antonio Giambruno,Sudarshan K. Sehgal Pdf

"Furnishes important research papers and results on group algebras and PI-algebras presented recently at the Conference on Methods in Ring Theory held in Levico Terme, Italy-familiarizing researchers with the latest topics, techniques, and methodologies encompassing contemporary algebra."

Author : Yuen Fong,U. Knauer
Publisher : De Gruyter Proceedings in Mathematics
Page : 372 pages
File Size : 49,9 Mb
Release : 1996
Category : Mathematics
ISBN : UOM:39015040706049

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First International Tainan-Moscow Algebra Workshop by Yuen Fong,U. Knauer Pdf

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