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 γ(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;α]] and the graph-theoretical properties of its directed zero-divisor graph γ(r[[x;α]]). in doing so, we give a characterization of the possible diameters of γ(r[[x;α]]) in terms of the diameter of γ(r), when the base ring r is reversible and right noetherian with an α-condition, namely α-compatible property. we also provide many examples for showing the necessity of our assumptions.