A generalization of purely extending modules relative to a torsion theory

In this work we introduce a new concept, namely, $\tau_{s}$-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 $R$ with unit, if $R$ is purely $\tau_{s}$-extending then every cyclic $\tau$-nonsingular $R$-module is flat and we show that this fact is true over a principal ideal domain as well. Also, we make a classification for the direct sums of the rings to be purely $\tau_{s}$-extending.