The annihilating-submodule graph of modules over commutative rings II

TL;DR

This paper studies the structure of the annihilating-submodule graph of modules over commutative rings, characterizing when it forms trees, stars, or paths, and relating graph properties to module structure.

Contribution

It characterizes the graph structures of AG(M) for modules over rings, linking graph topology to module decomposition and prime submodules.

Findings

If AG(M) is a tree, then it is either a star or a path of length 4.

For cyclic modules with at least three minimal prime submodules, the graph's girth is 3.

The chromatic number of AG(M) is at least the number of minimal prime submodules.

Abstract

Let M be a module over a commutative ring R. In this paper, we continue our study of annihilating-submodule graph AG(M) which was introduced in (The Zariski topology-graph of modules over commutative rings, Comm. Algebra., 42 (2014), 3283{3296). AG(M) is a (undirected) graph in which a nonzero submodule N of M is a vertex if and only if there exists a nonzero proper submodule K of M such that NK = (0), where NK, the product of N and K, is defined by (N : M)(K : M)M and two distinct vertices N and K are adjacent if and only if NK = (0). 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(AG(M))\geq |Min(M)|.