More on the Annihilator-Ideal Graph of a Commutative Ring

Abstract

Let $R$ be a commutative ring with identity and $\mathbb{A}(R)$ be the set of ideals of $R$ with non-zero annihilator. The annihilator-ideal graph of $R$, denoted by $A_I(R)$, is a simple graph with the vertex set $\mathbb{A}(R{)}^{\ast }:=\mathbb{A}(R)\setminus \{(0)\}$, and two distinct vertices $I$ and $J$ are adjacent if and only if ${\mathrm{Ann}}_R(IJ)\ne {\mathrm{Ann}}_R(I)\cup {\mathrm{Ann}}_R(J)$. In this paper, we study the affinity between the annihilator-ideal graph and the annihilating-ideal graph $\mathbb{AG}(R)$ (a well-known graph with the same vertices and two distinct vertices $I,J$ are adjacent if and only if $IJ=0$) associated with $R$. All rings whose $A_I(R)\ne \mathbb{AG}(R)$ and $\mathrm{gr}(A_I(R))=4$ are characterized. Among other results, we obtain necessary and sufficient conditions under which $A_I(R)$ is a star graph.