An extension of annihilating-ideal graph of commutative rings

Abstract

Let $R$ be a commutative ring with unity. The extension of annihilating-ideal graph of $R$, $\overline{\mathbb{AG}}(R)$, is the graph whose vertices are nonzero annihilating ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if there exist $n,m\in\mathbb{N}$ such that $I^{n}J^{m}=(0)$ with $I^{n}, J^{m}\neq (0)$. First, we differentiate when $\mathbb{AG}(R)$ and $\overline{\mathbb{AG}}(R)$ coincide. Then, we have characterized the diameter and the girth of $\overline{\mathbb{AG}}(R)$ when $R$ is a finite direct products of rings. Moreover, we show that $\overline{\mathbb{AG}}(R)$ contains a cycle, if $\overline{\mathbb{AG}}(R)\neq\mathbb{AG}(R)$.