Let R be a commutative ring with a non-zero identity and ? be an ideal R. In this paper, we introduce and investigate the concepts of ?-torsion modules, ?-torsion free modules, ?-flat modules and ?-von Neumann regular rings. Many examples, characterizations, and properties of these notions are given. Moreover, we use them to characterize reduced rings and ZN-rings.
Anebri et al. (Wed,) studied this question.