# Finitely Generated Annihilating-Ideal Graph of Commutative Rings

Document Type: Research Paper

Authors

1 Department of Mathematics, Shahrekord Branch, Islamic Azad Univercsity, Shahrekord, ‎Iran.

2 Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

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)).\$

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