Search

Let R be a ring and let S be a multiplicative subset of R. An R-module M is said to be a u-S-absolutely pure module if ExtR1N,M is u-S-torsion for any finitely presented R-module N. This paper introduces and studies the notion of S-FP-projective modules, which extends the classical notion of FP-projective modules. An R-module M is called an S-FP-projective module if ExtR1M,N=0 for any u-S-absolutely pure R-module N. We also introduce the S-FP-projective dimension of a module and the global S-FP-projective dimension of a ring. Then, the relationship between the S-FP-projective dimension and other homological dimensions is discussed.

In this paper, we characterize w-Noetherian rings and SM rings. More precisely, in terms of the u-operation on a commutative ring R, we prove that R is w-Noetherian if and only if the direct limit of rGV-torsion-free injective R-modules is injective and that R is SM, which can be regarded as a regular w-Noetherian ring, if and only if the direct limit of GV-torsion-free (or rGV-torsion-free) reg-injective R-modules is reg-injective. As a by-product of the proof of the second statement, we also obtain that the direct and inverse limits of u-modules are both u-modules and that SM rings are regular w-coherent.