Zero-divisor graphs of non-commutative rings

In a manner analogous to the commutative case, the zero-divisor graph of a non-commutative ring R can be defined as the directed graph Gamma(R) that its vertices are all non-zero zero-divisors of R in which for any two distinct vertices x and y, x -> y is an edge if and only if xy = 0. We investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of Gamma(R). In this paper it is shown that, with finitely many exceptions, if R is a ring and S is a finite semisimple ring which is not a field and Gamma(R)similar or equal to Gamma(S), then R similar or equal to S. For any finite field F and each integer n >= 2, we prove that if R is a ring and Gamma(R)similar or equal to Gamma(M-n (F)), then R similar or equal to M-n (F). Redmond defined the simple undirected graph Gamma(R) obtaining by deleting all directions on the edges in Gamma(R). We classify all ring R whose Gamma(R) is a complete graph, a bipartite graph or a tree. (c) 2005 Published by Elsevier Inc.