THE TOTAL TORSION ELEMENT GRAPH WITHOUT THE ZERO ELEMENT OF MODULES OVER COMMUTATIVE RINGS

Abstract

Abstract. Let M be a module over a commutative ring R, and let T(M)be its set of torsion elements. The total torsion element graph of Mover R is the graph T(Γ(M)) with vertices all elements of M, and twodistinct vertices m and n are adjacent if and only if m+ n ∈ T(M).In this paper, we study the basic properties and possible structures oftwo (induced) subgraphs Tor 0 (Γ(M)) and T 0 (Γ(M)) of T(Γ(M)), withvertices T(M) \{0}and M\{0}, respectively. The main purpose of thispaper is to extend the definitions and some results given in [6] to a moregeneral total torsion element graph case. 1. IntroductionThe concept of the graph of the zero-divisors of a ring was first introducedby Beck in [12] when discussing the coloring of a commutative ring. For thevertices of the graph, he takes all elements of a commutative ring R and twodistinct vertices a,b ∈ R are adjacent if ab = 0. There are many ways toassociate a graph to a given ring R. The most well-known is certainly thezero-divisor graph Γ(R) introduced in [9] whose vertices are the nonzero zero-divisors of R. Some properties of this graph may be found in [5] and [10]. In [8],Anderson and Badawi define, for a commutative ring R with nonzero identity,its total graph Γ(R). The set of vertices of this graph is R and two differentelements x,y ∈ R are adjacent if and only if x + y ∈ Z(R) which Z(R) is theset of all zero-divisors of R. For a recent generalization of this type of graphsee [7] and [11]. Let M be a module over a commutative ring R and let T(M)be the set of all torsion elements of M. In [14], the notion of the total torsionelement graph of a module over a commutative ring is introduced and denotedby T(Γ(M)), as the graph with all elements of M as vertices and for distinctm,n ∈ M, the vertices m and n are adjacent if and only if m+n ∈ T(M). Theycharacterize the girths and diameters of T(Γ(M)) and two (induced) subgraphs