%0 Journal Article
%T Directed zero-divisor graph and skew power series rings
%J Transactions on Combinatorics
%I University of Isfahan
%Z 2251-8657
%A Hashemi, Ebrahim
%A Yazdanfar, Marzieh
%A Alhevaz, Abdollah
%D 2018
%\ 12/01/2018
%V 7
%N 4
%P 43-57
%! Directed zero-divisor graph and skew power series rings
%K Zero-divisor graphs
%K Diameter
%K Reversible rings
%K Noetherian rings
%K Skew power series rings
%R 10.22108/toc.2018.109048.1543
%X 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.
%U https://toc.ui.ac.ir/article_23009_36558b6b0e7b2173c5ece6b5b1978c2e.pdf