On modules whose coclosed submodules are isomorphic to direct summands

Abstract

Inspired by the notion of virtually extending module, in this work we have proposed and investigated the behaviours of virtually lifting module as the dual of virtually extending module. Several properties of this structure have been discussed and got that virtually lifting module is a \(D_{12}\) module. It is observed that for a V ring R, a module M is virtually lifting iff it is lifting. Moreover, we have defined virtually \(D_{2}\) module as a proper extension of \(D_{2}\) module, and seen that virtually \(D_{2}\) modules is inherited by direct summands. Finally, we have investigated principally h-lifting module and proved that finite direct sum of principally h-lifting modules are principally h-lifting if each components are principally relatively projective.