FPn-Injective, FPn-Flat Covers and Preenvelopes, and Gorenstein AC-Flat Covers

We prove that, for any n ≥ 2, the classes of FPn-injective modules and of FPn-flat modules are both covering and preenveloping over any ring R. This includes the case of FP∞-injective and FP∞-flat modules (i.e., absolutely clean and, respectively, level modules). Then we consider a generalization of the class of (strongly) Gorenstein flat modules, i.e., the (strongly) Gorenstein AC-flat modules (cycles of exact complexes of flat modules that remain exact when tensored with any absolutely clean module). We prove that some of the properties of Gorenstein flat modules extend to the class of Gorenstein AC-flat modules; for example, we show that this class is precovering over any ring R. We also show that (as in the case of Gorenstein flat modules) every Gorenstein AC-flat module is a direct summand of a strongly Gorenstein AC-flat module. When R is such that the class of Gorenstein AC-flat modules is closed under extensions, the converse is also true. Moreover, we prove that if the class of Gorenstein AC-flat modules is closed under extensions, then it is covering.

(STRONGLY) GORENSTEIN FLAT MODULES OVER GROUP RINGS

Let $\Gamma $ be a group and ${\Gamma }^{\prime } $ be a subgroup of $\Gamma $ of finite index. Let $M$ be a $\Gamma $-module. It is shown that $M$ is (strongly) Gorenstein flat if and only if it is (strongly) Gorenstein flat as a ${\Gamma }^{\prime } $-module. We also provide some criteria in which the classes of Gorenstein projective and strongly Gorenstein flat $\Gamma $-modules are the same.

STRONGLY GORENSTEIN FLAT MODULES

In this paper, strongly Gorenstein flat modules are introduced and investigated. An R-module M is called strongly Gorenstein flat if there is an exact sequence ⋯→P1→P0→P0→P1→⋯ of projective R-modules with M=ker (P0→P1) such that Hom(−,F) leaves the sequence exact whenever F is a flat R-module. Several well-known classes of rings are characterized in terms of strongly Gorenstein flat modules. Some examples are given to show that strongly Gorenstein flat modules over coherent rings lie strictly between projective modules and Gorenstein flat modules. The strongly Gorenstein flat dimension and the existence of strongly Gorenstein flat precovers and pre-envelopes are also studied.