Inclusion graph of annihilators in a commutative ring

Abstract

Let $R$ be a commutative ring with identity, and let $Z(R)$ be the set of zero-divisors of $R$. The inclusion graph of annihilators in $R$, denoted by $Γ^{\prime}(R)$, is a graph with the vertex set $Z(R)^*=Z(R)\setminus\{0\}$ and two distinct vertices $x$ and $y$ are adjacent if and only if $\operatorname{ann}_R(x)\subseteq \operatorname{ann}_R(y)$ or $\operatorname{ann}_R(y)\subseteq \operatorname{ann}_R(x)$. It is proved that $Γ^{\prime}(R)$ is not connected if and only if $R$ is reduced with $|\operatorname{Min}(R)|=2$. Also, we show that if $Γ^{\prime}(R)$ is a connected graph, then the diameter of $Γ^{\prime}(R)$ is at most $4$ and the girth of $Γ^{\prime}(R)$ is at most $6$, if it contains a cycle. Moreover, we study the affinity between inclusion graph of annihilators and complement of the annihilator graph (a well-known graph with the same vertices and two distinct vertices $x$ and $y$ are adjacent if and only if $\operatorname{ann}_R(xy)\neq \operatorname{ann}_R(x)\cup \operatorname{ann}_R(y)$) associated with a commutative ring. Finally, we characterize all rings whose inclusion graphs of annihilators are complete.