Finitely Generated Annihilating-Ideal Graph of Commutative Rings

Abstract

Let $R$ be a commutative ring and $mathbb{A}(R)$ be the set of all ideals with non-zero annihilators. Assume that $mathbb{A}^*(R)=mathbb{A}(R)diagdown {0}$ and $mathbb{F}(R)$ denote the set of all finitely generated ideals of $R$. In this paper, we introduce and investigate the {it finitely generated subgraph} of the annihilating-ideal graph of $R$, denoted by $mathbb{AG}_F(R)$. It is the (undirected) graph with vertices $mathbb{A}_F(R)=mathbb{A}^*(R)cap mathbb{F}(R)$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. First, we study some basic properties of $mathbb{AG}_F(R)$. For instance, it is shown that if $R$ is not a domain, then $mathbb{AG}_F(R)$ has ascending chain condition (respectively, descending chain condition) on vertices if and only if $R$ is Noetherian (respectively, Artinian). We characterize all rings for which $mathbb{AG}_F(R)$ is a finite, complete, star or bipartite graph. Next, we study diameter and girth of $mathbb{AG}_F(R)$. It is proved that ${rm diam}(mathbb{AG}_F(R))leqslant {rm diam}(mathbb{AG}(R))$ and ${rm gr}(mathbb{AG}_F(R))={rm gr}(mathbb{AG}(R)).$