On Purities Relative to Minimal Right Ideals

We call a right module M weakly neat-flat if Hom(S, N) -> Hom(S, M) is surjective for any epimorphism N -> M and any simple right ideal S. A left module M is called weakly absolutely s-pure if S circle times M -> S circle times N is monic, for any monomorphism M -> N and any simple right ideal S. These notions are proper generalization of the neat-flat and the absolutely s-pure modules which are defined in the same way by considering all simple right modules of the ring, respectively. In this paper, we study some closure properties of weakly neat-flat and weakly absolutely s-pure modules, and investigate several classes of rings that are characterized via these modules. The relation between these modules and some well-known homological objects such as projective, flat, injective and absolutely pure are studied. For instance, it is proved that R is a right Kasch ring if and only if every weakly neat-flat right R-module is neat-flat (moreover if R is right min-coherent) if and only if every weakly absolutely s-pure left R-module is absolutely s-pure. The rings over which every weakly neat-flat (resp. weakly absolutely s-pure) module is injective and projective are exactly the QF rings. Finally, we study enveloping and covering properties of weakly neat-flat and weakly absolutely s-pure modules. The rings over which every simple right ideal has an epic projective envelope are characterized.