This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

A first course in noncommutative rings, by T. Y. Lam. Pp. 385. £37 (pb), £62.50 (hb). 2001. ISBN 0 387 95325 6 (pb), 0 387 95183 0 (hb) (Springer-Verlag).

Soient k un corps et A une k-algebre de dimension finie et de dimension globale ≤ 2. On construit une categorie triangulee C A associee a A, qui est triangle-equivalente a la categorie amassee C A si A est hereditaire. Lorsque C A est Hom-finie, on prouve qu'elle est 2-Calabi-Yau et munie d'un objet amas-basculant canonique. Cette nouvelle classe de categories contient certaines sous-categories stables de modules sur une algebre preprojective introduite par Geiss-Leclerc-Schroer et par Buan-Iyama-Reiten-Scott. Ces resultats s'appliquent aussi aux carquois a potentiel. Plus precisement, on introduit une categorie amassee C(Q,W) associee a un carquois a potentiel (Q,W). Quand il est Jacobi-fini, on prouve que cette categorie est munie d'un objet amas-basculant dont l'algebre d'endomorphismes est isomorphe a l'algebre jacobienne.

/pdf/cluster-categories-for-algebras-of-global-dimension-2-and-aj7oldygxl.pdf

Cluster categories for algebras of global dimension 2 and quivers with potential

Auslander-Reiten sequences and representation-finite algebras

On the derived category of a finite-dimensional algebra

The universal cover of a representation-finite algebra

Representation-finite algebras and multiplicative bases.

On algebras of strongly unbounded representation type

On Simply Connected Algebras

This paper deals with the representation theory of strongly locally finite quivers. We first study some properties of the finitely presented or copresented representations, and then construct in the category of locally finite dimensional representations some almost split sequences which start with a finitely co-presented representation and end with a finitely presented representation. Furthermore, we obtain a general description of the shapes of the Auslander-Reiten components of the category of finitely presented representations and prove that the number of regular Auslander-Reiten components is infinite if and only if the quiver is not of finite or infinite Dynkin type. In the infinite Dynkin case, we shall give a complete list of the indecomposable representations and an explicit description of the Auslander-Reiten components. Finally, we apply these results to study the Auslander-Reiten theory in the derived category of bounded complexes of finitely presented representations. 2010 Mathematics Subject Classification: 16G20; 16G70.

Representation theory of strongly locally finite quivers

This volume provides a systematic presentation of the theory of differential tensor algebras and their categories of modules It involves reduction techniques which have proved to be very useful in the development of representation theory of finite dimensional algebras The main results obtained with these methods are presented in an elementary and self contained way The authors provide a fresh point of view of well known facts on tame and wild differential tensor algebras, on tame and wild algebras, and on their modules But there are also some new results and some new proofs Their approach presents a formal alternative to the use of bocses (bimodules over categories with coalgebra structure) with underlying additive categories and pull-back reduction constructions Professional mathematicians working in representation theory and related fields, and graduate students interested in homological algebra will find much of interest in this book

/pdf/differential-tensor-algebras-and-their-module-categories-2y3uc6bso5.pdf

Raymundo Bautista

Papers

Representation-finite algebras and multiplicative bases.

On algebras of strongly unbounded representation type

On Simply Connected Algebras

Representation theory of strongly locally finite quivers

Differential Tensor Algebras and Their Module Categories