On the Nil R-mod Abelian Groups

  • Alireza NajafizadehEmail author
  • Fatemeh Karimi
  • Maryam Zafarkhah


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.


Nil Ring Torsion-free 

Mathematics Subject Classification (2010)




The authors express their sincere thanks to the referee for the valuable comments which highly improved the original manuscript.

Funding Information

This research was supported by a grant from Payame Noor University of I. R. of Iran. Moreover, this research received financial support from Payame Noor University during the preparation of this paper.


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Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Alireza Najafizadeh
    • 1
    Email author
  • Fatemeh Karimi
    • 1
  • Maryam Zafarkhah
    • 1
  1. 1.Department of MathematicsPayame Noor UniversityTehranIran

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