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 $Z_R(M)\setminus \text{Ann}_R(M)$, for $a,b \in R$ two distinct classes $[a]$ and $[b]$ are adjacent if and only if $\text{Ann}_M(a)\cap \text{Ann}_M(b)\neq 0$. 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 $Z_{R}(M)$ is an ideal of $R$. Also, we show that if $M$ is a finitely generated $R$-module with $r(\text{Ann}_R(M))\neq \text{Ann}_R(M)$ and $|m-\text{Ass}_R(M)|=1$ and $GA(M)$ is a star graph, then $r(\text{Ann}_{R}(M))$ is not a prime ideal of $R$ and $|V(GA(M))|=|\text{Min}\,\text{Ass}_R(M)|+1$.