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 \Gamma (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=۰ or yx=۰. It is well-known that for any commutative ring R, \Gamma (R) \cong \Gamma (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.

کلیدواژه ها:

Quotient ring ، zero-divisor graph ، reduced ring ، complemented graph