Torsion graph over multiplication modules

Abstract

For a commutative ring R, the torsion graph of an R-module M is 𝛤 (M) whose vertices are nonzero torsion elements of M, and two distinct vertices x and y are adjacent if and only if [x: M] [y: M ]M = 0. In this article we show that if S = R \ Z (M), then 𝛤 (M) and 𝛤 (S⁻¹M) are isomorphic for a multiplication R-module M. Also we prove that for a multiplication R-module M, if (𝛤 M) is uniquely complemented, then S⁻¹M is von Neumann regular or 𝛤 (M) is a star graph.