ON ZERO-DIVISOR GRAPHS OF QUOTIENT RINGS AND COMPLEMENTED ZERO-DIVISOR GRAPHS

For an arbitrary ring R, the zero-divisor graph of R, denoted by Γ(R), is an undirected simple graph that its vertices are all nonzero zero-divisors of R in which any two vertices x and y are adjacent if and only if either xy = 0 or yx = 0. It is well-known that for any commutative ring R, Γ(R) ∼= Γ(T(R)) where T(R) is the (total) quotient ring of R. In this paper we extend this fact for certain noncommutative rings, for example, reduced rings, right (left) self-injective rings and one-sided Artinian rings. The necessary and sufficient conditions for two reduced right Goldie rings to have isomorphic zero-divisor graphs is given. Also, we extend some known results about the zero-divisor graphs from the commutative to noncommutative setting: in particular, complemented and uniquely complemented graphs.