Quivers associated with finite rings - a cohomological approach

Abstract

In this paper we present the ongoing research on connection between digraphs
associated to finite (commutative) rings and quiver representations. Digraph
associated to a finite ring $A$ has the set of vertices $V = A^2$ and arrows
(or edges) $E=\left\{ \left( x,y\right)
\rightarrow\left( x+y,xy\right) \text{, }x,y\in A\right\}$. In another
terminology, it is a finite quiver with loops. In addition to previous work
to understand these graphs, the main goal of the present work is to
introduce some new cohomological and quiver methods. These methods
should provide us with better understanding of properties and classification of finite rings.