On the Nil R-mod Abelian Groups

Let M be an abelian group and N be a subgroup of M. Given a ring R, the notion of a left nil R-mod group modulo N is introduced as a generalization of a left nil R-mod group. Moreover, we define NilRl(M) as a subgroup of M which is defined to be the intersection of all subgroups N of M such that M is a left nil R-mod group modulo N. In this paper, we investigate some properties of the NilRl(M). Moreover, we describe it in some torsion-free abelian groups of rank two.