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.