Annihilating submodule graph for modules

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=۰‎. ‎In‎ ‎this paper we show that \Bbb G(M) is a connected graph‎, ‎{\rm‎ ‎diam}(\Bbb G(M))\leq ۳‎, ‎and {\rm gr}(\Bbb G(M))\leq ۴ 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 \{۰\} 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 \{۰\}‎. ‎Furthermore‎, ‎R‎ ‎is a field if and only if \Bbb G(M) is a complete graph‎, ‎for‎ ‎every M\in R-{\rm Mod}‎. ‎If R is a domain‎, ‎for every divisible‎ ‎module M\in R-{\rm Mod}‎, ‎\Bbb G(M) is a complete graph with‎ ‎\Bbb A^*(M)=\Bbb S(M)\setminus \{۰\}‎. ‎Among other things‎, ‎the‎ ‎properties of a reduced R-module M are investigated when‎ ‎\Bbb G(M) is a bipartite graph‎.

Keywords:

‎Module‎ , ‎Annihilating submodule graph‎ , ‎Complete graph