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 $\hat{I}{“}^{\prime }(R)$, is a graph with the vertex set $Z(R{)}^{\ast }=Z(R)\setminus \{0\}$ and two distinct vertices $x$ and $y$ are adjacent if and only if ${\mathrm{ann}}_R(x)\subseteq {\mathrm{ann}}_R(y)$ or ${\mathrm{ann}}_R(y)\subseteq {\mathrm{ann}}_R(x)$. It is proved that $\hat{I}{“}^{\prime }(R)$ is not connected if and only if $R$ is reduced with $|\mathrm{Min}(R)|=2$. Also, we show that if $\hat{I}{“}^{\prime }(R)$ is a connected graph, then the diameter of $\hat{I}{“}^{\prime }(R)$ is at most $4$ and the girth of $\hat{I}{“}^{\prime }(R)$ 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 ${\mathrm{ann}}_R(xy)\ne {\mathrm{ann}}_R(x)\cup {\mathrm{ann}}_R(y)$) associated with a commutative ring. Finally, we characterize all rings whose inclusion graphs of annihilators are complete.