Algebras, Rings and Modules
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.
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 proofIncludes the most important results in the theory of quasi-Frobenius and right serial ringsIn 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 ringsGives 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 chainGives the description of semiprime Noetherian semiperfect and semidistributive rings of injective dimension at most oneUses the technique of quivers to study of special classes of rings such as quasi-Frobenius rings and right serial ringsDiscusses the properties of global dimension of different classes of rings