IFP-Flat Dimensions and IFP-Injective Dimensions

In basic homological algebra, the flat and injective dimensions of modules play an important and fundamental role. In this paper, the closely related IFP-flat and IFP-injective dimensions are introduced and studied. We show that IFP-fd(M) = IFP-id(M+) and IFP-fd(M+)=IFP-id(M) for any R-module M over any ring R. Let (resp., ) be the class of all left (resp., right) R-modules of IFP-injective (resp., IFP-flat) dimension at most n. We prove that every right R-module has an - preenvelope, (,) is a perfect cotorsion theory over any ring R, and for any ring R with IFP-id(RR) ≤ n, (,) is a perfect cotorsion theory. This generalizes and improves the earlier work (J. Algebra 242 (2001), 447-459). Finally, some applications are given.