More on the Annihilator-Ideal Graph of a Commutative Ring

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 $\Bbb A®^{\ast} := \Bbb A ® \setminus\lbrace (0) \rbrace $, and two distinct vertices $I$ and $J$ are adjacent if and only if $$\mathrm{A}\mathrm{n}\mathrm{n}$ _{R} (IJ) \neq $\mathrm{A}\mathrm{n}\mathrm{n}$ _{R} (I) \cup $\mathrm{A}\mathrm{n}\mathrm{n}$ _{R} (J)$. In this paper, we study the affinity between the annihilator-ideal graph and the annihilating-ideal graph $\mathbb{A}\mathbb{G}(R)$ (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 $A_I(R)\mathrm{\ne }\mathbb{A}\mathbb{G}(R)$ and $\mathrm{g}\mathrm{r}(A_I(R))=4$ are characterized. Among other results, we obtain necessary and sufficient conditions under which $A_I(R)$ is a star graph.