The intersection graph of annihilator submodules of a module

Abstract

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 ZR(M) \ AnnR(M), for a, b ∈ R two distinct classes [a] and [b] are adjacent if and only if AnnM (a) ∩ AnnM (b) = 0. 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 r(AnnR(M)) = AnnR(M) and |m − AssR(M)| = 1 and GA(M) is a star graph, then r(AnnR(M)) is not a prime ideal of R and |V (GA(M))| = | Min AssR(M)| + 1.

On the module intersection graph of ideals of rings

“…Recently, there has been considerable attention in the literature to associating graphs with rings or modules. More specifically, there are many papers on assigning graphs to modules (see, for example, [3,11,13]). The present paper deals with what is known as the intersection graph.…”