A generalization of the total torsion element graph of modules over commutative rings

In this paper we present a new graph that is closely related to the total torsion element graph. Let M be a module over commutative ring R, Z(R) be the set of zero divisors of R and T(M) be the set of all torsionelements of M. The extended graph of a module M over commutative ring R is the graph with all elements of M as vertices and two nonzero distinct elements are adjacent if and only if there existsuch thatwith . The main purpose of this paper is to extend the definition and some results given in (Atani and Habibi, 2011).

Keywords:

Total torsion element graph , Zero divisor elements