A generalization of purely extending modules relative to a torsion theory

Abstract

In this work we introduce a new concept, namely, τsτs-extending modules (rings) which is torsion-theoretic analogues of extending modules and then we extend many results from extending modules to this new concept. For instance we show that for any ring RR with unit, if $_{R}R$ Ris purely τsτs-extending if and only if every cyclic ττ-nonsingular RR-module is flat. Also, we make a classification for the direct sums of the rings to be purely τsτs-extending.