The annihilating-ideal graph of $\mathbb{Z}_n$ is weakly perfect

Abstract

A graph is called weakly perfect if its vertex chromatic number equals its clique number. Let $R$ be a commutative ring with identity and $\mathbb{A}(R)$ be the set of ideals with non-zero annihilator. The annihilating-ideal graph of $R$ is defined as the graph $\mathbb{AG}(R)$ with the vertex set $\mathbb{A}(R)^{*}=\mathbb{A}(R)\setminus\{0\}$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=0$. In this paper, we show that the graph $\mathbb{AG}(\mathbb{Z}_n)$, for every positive integer $n$, is weakly perfect. Moreover, the exact value of the clique number of $\mathbb{AG}(\mathbb{Z}_n)$ is given and it is proved that $\mathbb{AG}(\mathbb{Z}_n)$ is class 1 for every positive integer ${n}$