On the torsion graph and von Neumann regular rings

Abstract

Let R be a commutative ring with identity and M be a unitary R-module. A torsion graph of M, denoted by Γ(M), is a graph whose vertices are the non-zero 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 paper, we investigate the relationship between the diameters of Γ(M) and Γ(R), and give some properties of minimal prime submodules of a multiplication R-module M over a von Neumann regular ring. In particular, we show that for a multiplication R-module M over a Bézout ring R the diameter of Γ(M) and Γ(R) is equal, where M T(M). Also, we prove that, for a faithful multiplication R-module M with |M| 4, Γ(M) is a complete graph if and only if Γ(R) is a complete graph.

Diameter and girth of Torsion Graph

“…Anderson and Badawi also introduced and investigated total graph of commutative ring in [1,2]. The zero-divisor graph of a commutative ring has been studied extensively by several authors [3,4,6,9,14,15,16]. The concept of zero-divisor graph has been extended to non-commutative rings by Redmond [17].…”

Planar torsion graph of modules

“…The concept of zero-divisor graph has been extended to non-commutative rings by Redmond [18], and has been extended to module by Ghalandarzadeh and Malakooti in [13]. The zero-divisor graph of a commutative ring and has been studied extensively by several authors [4,5,7,9,[14][15][16].…”

On the diameter of the graph ГAnn(M)(R)

“…For a reduced multiplication R-module M, they proved that, if Γ(M) is complemented, then S −1 M is von Neumann regular, where S = R \ Z(M). In addition, the authors in [16] have investigated the relationship between the diameter of Γ(M) and Γ(R).…”