Zero-divisor graph of matrix rings and Hurwitz rings

Let R be ring a with identity 1 not equal 0, S-n(R) be a subring of the ring T-n(R) of n x n upper triangular matrices over R, and H-n(R) be the ring defined in the next section using HR, the ring of the Hurwitz series over R. In this paper, we introduce the zero-divisor graph (Gamma) over right arrow (S-n(R)) and its underlying undirected graph Gamma(S-n(R)) of S-n(R). We give some basic graph theory properties of (Gamma) over right arrow (S-n(R)). Moreover, we obtain some results of the zero-divisor directed graph of (Gamma) over right arrow (H-n(R)).