Hashemi, E., Yazdanfar, M., Alhevaz, A. (2018). Directed zero-divisor graph and skew power series rings. Transactions on Combinatorics, 7(4), 43-57. doi: 10.22108/toc.2018.109048.1543

Ebrahim Hashemi; Marzieh Yazdanfar; Abdollah Alhevaz. "Directed zero-divisor graph and skew power series rings". Transactions on Combinatorics, 7, 4, 2018, 43-57. doi: 10.22108/toc.2018.109048.1543

Hashemi, E., Yazdanfar, M., Alhevaz, A. (2018). 'Directed zero-divisor graph and skew power series rings', Transactions on Combinatorics, 7(4), pp. 43-57. doi: 10.22108/toc.2018.109048.1543

Hashemi, E., Yazdanfar, M., Alhevaz, A. Directed zero-divisor graph and skew power series rings. Transactions on Combinatorics, 2018; 7(4): 43-57. doi: 10.22108/toc.2018.109048.1543

Directed zero-divisor graph and skew power series rings

^{}Department of Mathematics, Shahrood University of Technology, Shahrood, Iran

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 $x,y$, $x\rightarrow y$ 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.