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 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.
