Generalized $SV$-modules

Given an arbitrary quasiprojective right $R$-module $P$, we prove that every module in the category $\sigma(P)$ is weakly regular if and only if every module in $\sigma(M/I(M))$ is lifting, where $M$ is a generating object in $\sigma(P)$. In particular, we describe the rings over which every right module is weakly regular.