The intersection graph of annihilator submodules of a module

Let R be a commutative ring and M be a Noetherian R-module. The intersection graph of annihilator submodules of M, denoted by GA(M) is an undirected simple graph whose vertices are the classes of elements of [formula], for a, b ∈ R two distinct classes [a] and [b] are adjacent if and only if [formula]. In this paper, we study diameter and girth of GA(M) and characterize all modules that the intersection graph of annihilator submodules are connected. We prove that GA(M) is complete if and only if ZR{M) is an ideal of R. Also, we show that if M is a finitely generated R-module with [formula] and [formula] and GA(M) is a star graph, then r(AnnR(M)) is not a prime ideal of R and [formula].