Strongly Gorenstein flat modules and dimensions

Abstract

There is a variety of nice results about strongly Gorenstein flat modules over coherent rings. These results are done by Ding, Lie and Mao. The aim of this paper is to generalize some of these results, and to give homological descriptions of the strongly Gorenstein flat dimension (of modules and rings) over arbitrary associative rings.

Ding Projective Modules With Respect to a Semidualizing Module

“…Using the same methods, Mahdou and Tamekkante in [15] have proved the similar result holds for Ding projective modules. In the following, we will give a new proof to D C -projective modules.…”

Rings with finite Ding homological dimensions

“…, where all the A l s are Ding projective. By (♯) and [11,Theorem 2.4] it now also follows that C A n is Ding projective. With this, it is sufficient to prove the following: If P, A ∈ C = (R) are complexes of, respectively, projective and Ding projective modules, and P ≃ X ≃ A , then the cokernel C P n is Ding projective if and only if C A n is so.…”

“…By [11,Theorem 2.4] we have Ext g−s R (C A n , Q) ̸ = 0 for some flat R -module Q , from which H −g (RHom R (X, Q)) ̸ = 0 by (♯) and ( * ) follows. We conclude that n ≥ supX .…”

(Strongly) Gorenstein Flat Modules Over Group rings

“…R 3.2. We should remark that, as in [11,19], we cannot find an example of a Gorenstein projective module which is not strongly Gorenstein flat. It follows from [9, Theorem 4.2] that the projective dimension of any flat Γ-module is less than or equal to one, whenever Γ is a finite group.…”

“…In addition, it is proved in [11] that over coherent rings, strongly Gorenstein flat modules lie strictly between projective and Gorenstein flat modules. Strongly Gorenstein flat modules and dimensions were studied further in [19,20]. In this note, we study (strongly) Gorenstein flat modules over group rings.…”