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 g(m)‎, ‎is an‎ ‎undirected graph with vertex set a∗(m) and two distinct‎ ‎elements n and k of a∗(m) are adjacent if n^∗k=0‎. ‎in‎ ‎this paper we show that g(m) is a connected graph‎, ‎‎‎diam(g(m))≤3‎, ‎and gr(g(m))≤4 if ‎‎g(m) contains a cycle‎. ‎moreover‎, ‎g(m) is an empty graph‎ ‎if and only if ann(m) is a prime ideal of r and ‎‎a^∗(m)≠s(m)∖{0} if and only if m is a‎ ‎uniform r-module‎, ‎ann(m) is a semi-prime ideal of r‎ ‎and a∗(m)≠s(m)∖{0}‎. ‎furthermore‎, ‎r‎ ‎is a field if and only if g(m) is a complete graph‎, ‎for‎ ‎every m∈r−mod‎. ‎if r is a domain‎, ‎for every divisible‎ ‎module m∈r−mod‎, ‎g(m) is a complete graph with‎ ‎a^∗(m)=s(m)∖{0}‎. ‎among other things‎, ‎the‎ ‎properties of a reduced r-module m are investigated when‎ ‎g(m) is a bipartite graph‎.