The extended zero-divisor graph of a commutative ring I

Abstract

Let $R$ be a commutative ring with identity, and let $Z(R)$ be the set of zero-divisors of $R$. The extended zero-divisor graph of $R$ is the undirected (simple) graph $\Gamma'(R)$ with the vertex set $Z(R)^*=Z(R)\setminus\{0\}$, and two distinct vertices $x$ and $y$ are adjacent if and only if either $Rx\cap \mathrm{Ann}(y)\neq (0)$ or $Ry\cap \mathrm{Ann}(x)\neq (0)$. It follows that the zero-divisor graph $\Gamma(R)$ is a subgraph of $\Gamma'(R)$. It is proved that $\Gamma'(R)$ is connected with diameter at most two and with girth at most four, if $\Gamma'(R)$ contains a cycle. Moreover, we characterize all rings whose extended zero-divisor graphs are complete or star. Furthermore, we study the affinity between extended zero-divisor graph and zero-divisor graph associated with a commutative ring. For instance, for a non-reduced ring $R$, it is proved that the extended zero-divisor graph and the zero-divisor graph of $R$ are identical to the join of a complete graph and a null graph if and only if $ann_R(Z(R))$ is a prime ideal.