J. Korean Math. Soc. 51 (2014), No. 4, pp. 721–734 http://dx.doi.org/10.4134/JKMS.2014.51.4.721 THE TOTAL TORSION ELEMENT GRAPH WITHOUT THE ZERO ELEMENT OF MODULES OVER COMMUTATIVE RINGS Fatemeh Esmaeili Khalil Saraei 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 M over R is the graph T (Γ(M)) with vertices all elements of M, and two distinct 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 of two (induced) subgraphs T or 0 (Γ(M)) and T 0 (Γ(M)) of T (Γ(M)), with vertices T (M) \{0} and M \{0}, respectively. The main purpose of this paper is to extend the definitions and some results given in [6] to a more general total torsion element graph case. 1. Introduction The concept of the graph of the zero-divisors of a ring was first introduced by Beck in [12] when discussing the coloring of a commutative ring. For the vertices of the graph, he takes all elements of a commutative ring R and two distinct vertices a, b ∈ R are adjacent if ab = 0. There are many ways to associate a graph to a given ring R. The most well-known is certainly the zero-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 different elements x, y ∈ R are adjacent if and only if x + y ∈ Z (R) which Z (R) is the set of all zero-divisors of R. For a recent generalization of this type of graph see [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 torsion element graph of a module over a commutative ring is introduced and denoted by T (Γ(M )), as the graph with all elements of M as vertices and for distinct m, n ∈ M , the vertices m and n are adjacent if and only if m+ n ∈ T (M ). They characterize the girths and diameters of T (Γ(M )) and two (induced) subgraphs Received July 27, 2013; Revised October 31, 2013. 2010 Mathematics Subject Classification. Primary 05C99, 13C13. Key words and phrases. total graph, torsion prime submodule, T -reduced. c 2014 Korean Mathematical Society 721