On the Nil R-mod Abelian Groups

Abstract

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 \(\text {Nil}^{l}_{R}(M)\) 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 \(\text {Nil}^{l}_{R}(M)\). Moreover, we describe it in some torsion-free abelian groups of rank two.