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Algebras, Rings and Modules: Volume 1: Mathematics and Its Applications, cartea 575

Autor Michiel Hazewinkel, Nadiya Gubareni, V.V. Kirichenko
en Limba Engleză Paperback – 29 ian 2011
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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Specificații

ISBN-13: 9789048167043
ISBN-10: 9048167043
Pagini: 392
Ilustrații: XII, 380 p.
Dimensiuni: 160 x 240 x 21 mm
Greutate: 0.55 kg
Ediția:2004
Editura: SPRINGER NETHERLANDS
Colecția Springer
Seria Mathematics and Its Applications

Locul publicării:Dordrecht, Netherlands

Public țintă

Research

Cuprins

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.

Recenzii

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)

Caracteristici

Contains almost all of the most important results in the theory of representations of posets, quivers and their applications to the representations of finite groups and finite dimensional algebras, most of them are given with full proof Includes the most important results in the theory of quasi-Frobenius and right serial rings In particular, gives the algorithm of the construction of a finite Frobenius ring by means of a finite partially ordered set and it is provides the proof of the Cartan Determinant Conjecture for right serial Artinian rings Gives the full description of semiprime Noetherian semiperfect and semidistributive rings. In particular, with any finite poset it may be associated a prime Noetherian semiperfect and semidistributive ring with nonzero Jacobson radical and a finite ergodic Markov chain Gives the description of semiprime Noetherian semiperfect and semidistributive rings of injective dimension at most one Uses the technique of quivers to study of special classes of rings such as quasi-Frobenius rings and right serial rings Discusses the properties of global dimension of different classes of rings