Torsion graph of modules

Abstract

Let R be a commutative ring and M be an R-module. We associate to M a graph denoted by, Γ(M) called the torsion graph of M, whose vertices are the non-zero torsion elements of M and two distinct elements x; y are adjacent if and only if [x : M][y : M]M = 0. We investigate the interplay between module-theoretic properties of M and graph-theoretic properties of Γ(M). Among other results, we prove that Γ(M) is connected and diam(Γ(M)) ≤ 3 for a faithful R-module M.