Central Sets in the Annihilating-Ideal Graph of Commutative Rings

Abstract

Let $R$ be a commutative ring with identity and ${\mathbb{A}}^{\ast }(R)$ be the set of non-zero ideals with non-zero annihilators. The annihilating-ideal graph of $R$ is defined as the graph $\mathbb{A}\mathbb{G}(R)$ with the vertex set ${\mathbb{A}}^{\ast }(R)$ and two distinct vertices $I_1$ and $I_2$ are adjacent if and only if $I_1{I}_2=(0)$. In this paper, we study some connections between the graph-theoretic properties of $\mathbb{A}\mathbb{G}(R)$ and algebraic properties of the commutative ring $R$.