Remarks on fp-Injective and fp-Flat Modules

A left R-module M is said to be fp-injective if, for every monomorphism K -> L with K and L finitely presented left R-modules, Hom(L, M) -> Hom(K, M) is an epimorphism. A right R-module N is called fp-flat if, for every monomorphism K -> L with K and L finitely presented left R-modules, N circle times K -> N circle times L is a monomorphism. In this note, we study precovers and preenvelopes by fp-injective and fp-flat modules, including their properties under (almost) excellent extensions of rings. In addition, we also introduce and investigate fp-projective modules.