Rings over which every module has a flat $\delta$-cover

Let $M$ be a module. A {\em $\delta$-cover} of $M$ is an epimorphism from a module $F$ onto $M$ with a $\delta$-small kernel. A $\delta$-cover is said to be a {\em flat $\delta$-cover} in case $F$ is a flat module. In the present paper, we investigate some properties of (flat) $\delta$-covers and flat modules having a projective $\delta$-cover. Moreover, we study rings over which every module has a flat $\delta$-cover and call them {\em right generalized $\delta$-perfect} rings. We also give some characterizations of $\delta$-semiperfect and $\delta$-perfect rings in terms of locally (finitely, quasi-, direct-) projective $\delta$-covers and flat $\delta$-covers.