Directed zero-divisor graph and skew power series rings

Abstract

‎Let $R$ be an associative ring with identity and $Z^{ast}(R)$ 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 $Gamma{(R)}$‎, ‎is the directed graph whose vertices are the set of non-zero zero-divisors of $R$ and for distinct non-zero zero-divisors $x,y$‎, ‎$xrightarrow y$ 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 $R[[x;alpha]]$ and the graph-theoretical properties of its directed zero-divisor graph $Gamma(R[[x;alpha]])$‎. ‎In doing so‎, ‎we give a characterization of the possible diameters of $Gamma(R[[x;alpha]])$ in terms of the diameter of $Gamma(R)$‎, ‎when the base ring $R$ is reversible and right Noetherian with an‎ ‎$alpha$-condition‎, ‎namely $alpha$-compatible property‎. ‎We also provide many examples for showing the necessity of our assumptions‎.