Annihilating submodule graph for modules

Abstract

Let $R$ be a commutative ring and $M$ an‎ ‎$R$-module‎. ‎In this article‎, ‎we introduce a new generalization of‎ ‎the annihilating-ideal graph of commutative rings to modules‎. ‎The‎ ‎annihilating submodule graph of $M$‎, ‎denoted by $Bbb G(M)$‎, ‎is an‎ ‎undirected graph with vertex set $Bbb A^*(M)$ and two distinct‎ ‎elements $N$ and $K$ of $Bbb A^*(M)$ are adjacent if $N*K=0$‎. ‎In‎ ‎this paper we show that $Bbb G(M)$ is a connected graph‎, ‎${rm‎ ‎diam}(Bbb G(M))leq 3$‎, ‎and ${rm gr}(Bbb G(M))leq 4$ if $Bbb‎ ‎G(M)$ contains a cycle‎. ‎Moreover‎, ‎$Bbb G(M)$ is an empty graph‎ ‎if and only if ${rm ann}(M)$ is a prime ideal of $R$ and $Bbb‎ ‎A^*(M)neq Bbb S(M)setminus {0}$ if and only if $M$ is a‎ ‎uniform $R$-module‎, ‎${rm ann}(M)$ is a semi-prime ideal of $R$‎ ‎and $Bbb A^*(M)neq Bbb S(M)setminus {0}$‎. ‎Furthermore‎, ‎$R$‎ ‎is a field if and only if $Bbb G(M)$ is a complete graph‎, ‎for‎ ‎every $Min R-{rm Mod}$‎. ‎If $R$ is a domain‎, ‎for every divisible‎ ‎module $Min R-{rm Mod}$‎, ‎$Bbb G(M)$ is a complete graph with‎ ‎$Bbb A^*(M)=Bbb S(M)setminus {0}$‎. ‎Among other things‎, ‎the‎ ‎properties of a reduced $R$-module $M$ are investigated when‎ ‎$Bbb G(M)$ is a bipartite graph‎.