The annihilating-submodule graph of modules over commutative rings II

Abstract

Let M be a module over a commutative ring R. The annihilating-submodule graph of M, denoted by AG(M), is a simple graph in which a non-zero submodule N of M is a vertex if and only if there exists a non-zero proper submodule K of M such that N K = (0), where N K, the product of N and K, is denoted by (N : M)(K : M)M and two distinct vertices N and K are adjacent if and only if N K = (0). This graph is a submodule version of the annihilating-ideal graph. We prove that if AG(M) is a tree, then either AG(M) is a star graph or a path of order 4 and in the latter case $${M\cong F \times S}$$ , where F is a simple module and S is a module with a unique non-trivial submodule. Moreover, we prove that if M is a cyclic module with at least three minimal prime submodules, then gr(AG(M)) = 3 and for every cyclic module M, $${cl({\rm AG}(M)) \geq |{\rm Min}(M)|}$$ .