On quasi-zero divisor graphs of non-commutative rings

Document Type : Research Paper


Faculty of Mathematical sciences, Shahrood University of Technology, Shahrood, Iran.



Let $R$ be an associative ring with identity. A ring $R$ is called reversible if $ab=0$, then $ba=0$ for $a,b\in R$.
The quasi-zero-divisor graph of $R$, denoted by $\Gamma^*(R)$ is an undirected graph with all nonzero zero-divisors of $R$ as vertex set and two distinct vertices $x$ and $y$ are adjacent if and only if there exists $0\neq r\in R \setminus (\mathrm{ann}(x) \cup \mathrm{ann}(y))$ such that $xry=0$ or $yrx=0$. In this paper, we determine the diameter and girth of $\Gamma^*(R)$. We show that  the  zero-divisor graph of $R$ denoted by $\Gamma(R)$, is an induced subgraph of $\Gamma^*(R)$. Also, we investigate when $\Gamma^*(R)$ is identical to $\Gamma(R)$. Moreover, for a reversible ring $R$, we study the diameter and girth of $\Gamma^*(R[x])$ and we investigate when $\Gamma^*(R[x])$ is identical to $\Gamma(R[x])$.


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