Diameter and girth of Torsion Graph

Abstract

Let R be a commutative ring with identity. Let M be an R-module and T (M )

Compressed Intersection Annihilator Graph

“…Also, we show that the compressed intersection annihilator graph will never be a complete bipartite graph. In the third section, we generalize some results from [16]. In addition, we show that the graph IA(R) with at least three vertices is connected and its diameter is less than or equal to three.…”

“…In this section, we determine the diameter of the graph and its girth. Now, the following theorems 3.1, 3.2 and corollary 3.3 are generalizations of theorems 3.1, 4.1 and corollary 4.2 from [16] respectively, if we consider R as an R-module in the graph Γ R (R). Theorem 3.1 shows that if either the ring R is a von Neumann regular ring and R ≇ ann R (x) ⊕ ann R (y) for any two distinct x, y ∈ Z * (R) or Nil(R) = {0}, then IA(R) is connected with a diameter less than or equal to three.…”

“…The proofs of the following theorems 3.1, 3.2 and corollary 3.3 are analogues to the proofs of parts 2, 3 of theorem 3.1, theorem 4.1 and corollary 4.2 of [16] respectively. We have just to replace T (M) * by Z(R/ ∼) * , Ann(x) by ann R (x) and the vertices in Γ R (M) by the vertices in IA(R).…”

“…P. Malakooti Rad, S. Yassemi, Sh. Ghalandarzadeh and P. Safari in [16], with an Rmodule M. The set of vertices of Γ R (M) is the set of non-zero torsion elements T (M) * where, T (M) * = {m ∈ M | ann R (m) = {0}} with two distinct vertices x and y are adjacent if and only if ann R (x) ∩ ann R (y) = {0}. In their work, they studied in which case Γ R (M) is connected with diam(Γ R (M)) is less than or equal to three, the relationship between the diameter of Γ R (M) and Γ R (R) and proved that the girth of Γ R (M) belongs to {3, ∞}.…”

Planar torsion graph of modules

“…Let R be a commutative ring with identity and M be a unitary R-module. In this paper, we investigate the concept of torsion-graph for module that was introduced by Malakooti and Yassemi in [17]. Here the torsion graph Γ R (M) of M is a simple graph whose vertices are non-zero torsion elements of M and two different elements x, y are adjacent if and only if Ann(x) ∩ Ann(y) 0.…”