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

Abstract

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 F P∞-injective and F P∞-flat modules (i.e. absolutely clean and, respectively, level modules). Then we consider a generalization of the class of (strongly) Gorenstein flat modules -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 ACflat module. When R is such that the class of Gorenstein AC-flat modules is closed under extensions, the converse is also true. We also prove that if the class of Gorenstein AC-flat modules is closed under extensions, then this class of modules is covering.

Locally type $\text{FP}_n$ and $n$-coherent categories

“…For example, see [28,Corollary 2.3.7] or [49, Section 2.2]. 8 Proof. It is only left to prove the second statement.…”

Gorenstein flat modules with respect to duality pairs

“…Hence, by the middle column and Lemma 2.10, W is in GF (X ,Y) (R). Now, in the short exact sequence 0 → G → W → M → 0, we note that G and W are in GF (X ,Y) (R), and Tor R i (Y, M ) = 0 for all i > 0 and all Y ∈ Y, then M is in GF (X ,Y) (R) by (2).…”

“…(2) If X = F , then Gorenstein (X , Y)-flat modules are exactly Gorenstein Y-flat modules GF Y (R) in [7]. If Y is the class of all absolutely clean modules, then Gorenstein Y-flat modules are precisely Gorenstein AC-flat modules in [2].…”

“…Later, many scholars further studied these modules and introduced various generalizations of these modules (See, e.g., [1,3,4,14,15,19]). For example, Bravo, Estrada and Iacob defined Gorenstein AC-flat modules in [2], Estrada, Iacob and Péres studied Gorenstein B-flat modules in [7], where B is a class of right R-modules.…”

“…

Let X be a class of left R-modules, Y be a class of right R-modules. In this paper, we introduce and study Gorenstein (X , Y)-flat modules as a common generalization of some known modules such as Gorenstein flat modules [9], Gorenstein n-flat modules [22], Gorenstein B-flat modules [7], Gorenstein AC-flat modules [2], Ω-Gorenstein flat modules [10] and so on. We show that the class of all Gorenstein (X , Y)-flat modules have a strong stability.

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