Finitely Generated Annihilating-Ideal Graph of Commutative Rings

Let r be a commutative ring and a(r) be the set of all ideals with non-zero annihilators. assume that a*(r) = a(r){(0)} and f(r) denote the set of all finitely generated ideals of r. in this paper, we introduce and investigate the finitely generated annihilating-ideal graph of r, denoted by agf (r). it is the (undirected) graph with vertices af (r) = a* (r) ∩ f(r) and two distinct vertices i and j are adjacent if and only if = (0). first, we study some basic properties of agf(r). for instance, it is shown that if r is not a domain, then agf (r) has ascending chain condition on vertices if and only if r is noetherian. we characterize all rings for which agf (r) is a finite, complete, star or bipartite graph. next, we study diameter and girth of agf(r). it is proved that diam(agf(r)) ≤ diam(ag(r)) and gr(agf(r)) = gr(ag(r)).