Neat-flat Modules

Let R be a ring. A right R- module M is said to be neat- flat if the kernel of any epimorphism Y -> M is neat in Y, i. e., the induced map Hom(S, Y) -> Hom(S, M) is surjective for any simple right R-module S. Neat-flat right R-modules are projective if and only if R is a right Sigma-CS ring. Every cyclic neat-flat right R-module is projective if and only if R is right CS and right C-ring. It is shown that, over a commutative Noetherian ring R, (1) every neat-flat module is flat if and only if every absolutely coneat module is injective if and only if R congruent to A x B, wherein A is a QF-ring and B is hereditary, and (2) every neat-flat module is absolutely coneat if and only if every absolutely coneat module is neat-flat if and only if R congruent to A x B, wherein A is a QF-ring and B is Artinian with J(2)(B) = 0.