Overview
- 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
Part of the book series: Mathematics and Its Applications (MAIA, volume 586)
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Table of contents (7 chapters)
Keywords
About this book
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.
Reviews
From the reviews:
"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)
"The authors concentrate on the representation theory (of groups, algebras, posets …) and on the structure of some special rings. … The book is written on a level accessible to advanced students who have some experience with algebra. … Interesting historical comments and references close each chapter. A list of references for further reading is providing at the end of the book." (Stanislaw Kasjan, Mathematical Reviews, Issue 2009 b)
Editors and Affiliations
Bibliographic Information
Book Title: Algebras, Rings and Modules
Book Subtitle: Volume 2
Editors: Michiel Hazewinkel, Nadiya Gubareni, V.V. Kirichenko
Series Title: Mathematics and Its Applications
DOI: https://doi.org/10.1007/978-1-4020-5141-8
Publisher: Springer Dordrecht
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Science+Business Media B.V. 2007
Hardcover ISBN: 978-1-4020-5140-1Published: 12 September 2007
Softcover ISBN: 978-90-481-7292-4Published: 30 November 2010
eBook ISBN: 978-1-4020-5141-8Published: 08 September 2007
Edition Number: 1
Number of Pages: XII, 400
Topics: Associative Rings and Algebras, Category Theory, Homological Algebra, Commutative Rings and Algebras, Linear and Multilinear Algebras, Matrix Theory