DIRECTED ZERO-DIVISOR GRAPH an‎d SKEW POWER SERIES RINGS

Let R be an associative ring with identity and Z (R) be its set of non-zero zero-divisors. Zero-divisor graphs of rings are well represented in the literature of commutative and non-commutative rings. The directed zero-divisor graph of R, denoted by (R), is the directed graph whose vertices are the set of non-zero zero-divisors of R and for distinct non-zero zero-divisors x; y, x ! y is an directed edge if and only if xy = 0. In this paper, we connect some graph-theoretic concepts with algebraic notions, and investigate the interplay between the ring-theoretical properties of a skew power series ring R[x; ] and the graph-theoretical properties of its directed zero-divisor graph (R[x; ]). In doing so, we give a characterization of the possible diameters of (R[x; ]) in terms of the diameter of (R), when the base ring R is reversible and right Noetherian with an -condition, namely -compatible property. We also provide many examples for showing the necessity of our assumptions.