R. Taheri - Department of Mathematics, Shahrekord Branch, Islamic Azad Univercsity, Shahrekord, ‎Iran.
A. Tehranien - Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

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 {۰} 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=(۰). 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)).

Commutative rings, Annihilating-ideal, Finitely generated ideal, Graph