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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.
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This text brings the reader to the frontiers of current research in topological rings. The exercises illustrate many results and theorems while a comprehensive bibliography is also included. The book is aimed at those readers acquainted with some very basic point-set topology and algebra, as normally presented in semester courses at the beginning graduate level or even at the advanced undergraduate level. Familiarity with Hausdorff, metric, compact and locally compact spaces and basic properties of continuous functions, also with groups, rings, fields, vector spaces and modules, and with Zor
Topological rings. --- Rings, Topological --- Associative rings --- Commutative rings
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This beautiful book is the result of the author's wide and deep knowledge of the subject matter, combined with a gift for exposition ... The well selected material is offered in an integrated presentation of the structure theory of noncommutative (associative rings) and its applications. The book will appeal to many a reader. It would be wonderful as a textbook, and in fact, it's based on the author's lecture notes ... Only people looking for the most general form of a particular theorem are advised to turn to other books, but those interested in studying or reviewing its subject matter or looking for a rounded account of it, could do no better than choosing this book for this purpose. Bulletin of the American Mathematical Society Noncommutative Rings provides a cross-section of ideas, techniques and results that give the reader an idea of that part of algebra which concerns itself with noncommutative rings. In the space of 200 pages, Herstein covers the Jacobson radical, semisimple rings, commutativity theorems, simple algebras, representations of finite groups, polynomial identities, Goldie's theorem and the Golod-Shafarevitch theorem. Almost every practicing ring theorist has studied portions of this classic monograph.
Noncommutative rings --- Non-commutative rings --- Associative rings --- Noncommutative rings.
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The study of noncommutative rings is a major area in modern algebra. The structure theory of noncommutative rings was originally concerned with three parts: The study of semi-simple rings; the study of radical rings; and the construction of rings with given radical and semi-simple factor rings. Recently, this has extended to many new parts: The zero-divisor theory, containing the study of coefficients of zero-dividing polynomials and the study of annihilators over noncommutative rings, that is related to the Köthe's conjecture; the study of nil rings and Jacobson rings; the study of applying r
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A broad range of topics is covered here, including commutative monoid rings, the Jacobson radical of semigroup rings, blocks of modular group algebras, nilpotency index of the radical of group algebras, the isomorphism problem for group rings, inverse semigroup algebras and the Picard group of an abelian group ring. The survey lectures provide an up-to-date account of the current state of the subject and form a comprehensive introduction for intending researchers.
Group rings --- Semigroup rings --- Congresses
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This 2004 introduction to noncommutative noetherian rings is intended to be accessible to anyone with a basic background in abstract algebra. It can be used as a second-year graduate text, or as a self-contained reference. Extensive explanatory discussion is given, and exercises are integrated throughout. Various important settings, such as group algebras, Lie algebras, and quantum groups, are sketched at the outset to describe typical problems and provide motivation. The text then develops and illustrates the standard ingredients of the theory: e.g., skew polynomial rings, rings of fractions, bimodules, Krull dimension, linked prime ideals. Recurring emphasis is placed on prime ideals, which play a central role in applications to representation theory. This edition incorporates substantial revisions, particularly in the first third of the book, where the presentation has been changed to increase accessibility and topicality. Material includes the basic types of quantum groups, which then serve as test cases for the theory developed.
Noetherian rings. --- Noncommutative rings. --- Non-commutative rings --- Associative rings --- Rings, Noetherian --- Commutative rings
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Graded Ring Theory
Ordered algebraic structures --- Graded rings. --- Noncommutative rings. --- Graded rings --- Noncommutative rings --- 512.55 --- Rings (Algebra) --- 512.55 Rings and modules --- Rings and modules --- Non-commutative rings --- Associative rings
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Rings That are Nearly Associative
Associative rings. --- Rings (Algebra) --- Algebraic rings --- Ring theory --- Algebraic fields
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This monograph first published in 1986 is a reasonably self-contained account of a large part of the theory of non-commutative Noetherian rings. The author focuses on two important aspects: localization and the structure of infective modules. The former is presented in the opening chapters after which some new module-theoretic concepts and methods are used to formulate a new view of localization. This view, which is one of the book's highlights, shows that the study of localization is inextricably linked to the study of certain injectives and leads, for the first time, to some genuine applications of localization in the study of Noetherian rings. In the last part Professor Jategaonkar introduces a unified setting for four intensively studied classes of Noetherian rings: HNP rings, PI rings, enveloping algebras of solvable Lie algebras, and group rings of polycyclic groups. Some appendices summarize relevant background information about these four classes.
Noetherian rings. --- Localization theory. --- Categories (Mathematics) --- Homotopy theory --- Nilpotent groups --- Rings, Noetherian --- Associative rings --- Commutative rings
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This monograph is concerned with exchange rings in various conditions related to stable range. Diagonal reduction of regular matrices and cleanness of square matrices are also discussed. Readers will come across various topics: cancellation of modules, comparability of modules, cleanness, monoid theory, matrix theory, K-theory, topology, amongst others. This is a first-ever book that contains many of these topics considered under stable range conditions. It will be of great interest to researchers and graduate students involved in ring and module theories.
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