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Algebras, Rings and Modules: Volume 1 / Edition 1

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Overview

As a natural continuation of the first volume of Algebras, Rings and Modules, this book provides both the classical aspects of the theory of groups and their representations as well as a general introduction to the modern theory of representations including the representations of quivers and finite partially ordered sets and their applications to finite dimensional algebras.

Detailed attention is given to special classes of algebras and rings including Frobenius, quasi-Frobenius, right serial rings and tiled orders using the technique of quivers. The most important recent developments in the theory of these rings are examined.

The Cartan Determinant Conjecture and some properties of global dimensions of different classes of rings are also given. The last chapters of this volume provide the theory of semiprime Noetherian semiperfect and semidistributive rings.

Of course, this book is mainly aimed at researchers in the theory of rings and algebras but graduate and postgraduate students, especially those using algebraic techniques, should also find this book of interest.

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Editorial Reviews

From the Publisher

From the reviews of the first edition:

"This is the first of two volumes which aim to take the theory of associative rings and their modules from fundamental definitions to the research frontier. The book is written at a level intended to be accessible to students who have taken standard basic undergraduate courses in linear algebra and abstract algebra. … has been written with considerable attention to accuracy, and has been proofread with care. … A very welcome feature is the substantial set of bibliographic and historical notes at the end of each chapter." (Kenneth A. Brown, Mathematical Reviews, 2006a)

"This book follows in the footsteps of the valuable work done during the seventies of systematizing the investigation of properties and structure of rings by using their categories of modules. … A remarkable novelty in the present monograph is the study of semiperfect rings by means of quivers. … Another good idea is the inclusion of the study of commutative as well as non-commutative discrete valuation rings. Each chapter ends with some illustrative historical notes." (José Gómez Torrecillas, Zentralblatt MATH, Vol. 1086 (12), 2006)

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Product Details

  • ISBN-13: 9789048167043
  • Publisher: Springer Netherlands
  • Publication date: 1/11/2011
  • Series: Mathematics and Its Applications (closed) Series , #575
  • Edition description: Softcover reprint of hardcover 1st ed. 2004
  • Edition number: 1
  • Pages: 380
  • Product dimensions: 6.14 (w) x 9.21 (h) x 0.81 (d)

Table of Contents

Preliminaries.- Decompositions of rings.- Artinian and Noetherian rings.- Categories and functors.- Projectives, injectives and flats.- Homological dimensions.- Integral domains.- Dedekind domains.- Goldie rings.- Semiperfect rings.- Quivers of rings.- Serial rings and modules.- Serial rings and their properties.- Semiperfect semidistributive rings.

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