n-Flat and n-FP-injective modules

$n$-strongly Gorenstein graded modules

“…in gr-R with each F + i gr-injective by [26], Lemma 4.1. For any X ∈ gr-R and N ∈ R-gr, we have EXT 1 R (N, X + ) ∼ = Tor R 1 (X, N ) + by [17], Lemma 2.1.…”

“…In [17], García Rozas et al proved the existence of flat covers in the category of graded modules over a graded ring. Also, the homological properties of FP-gr-injective modules over a gr-coherent ring were investigated in [4], [26]. On the other hand, Asensio, López-Ramos and Torrecillas in [1], [2] introduced the notions of Gorenstein gr-projective, gr-injective and gr-flat modules.…”

Category of $n$-FCP-gr-projective modules with respect to special copresented graded modules

“…Several authors have investegated the graded aspect of some notions in relative homological algerbra. For example, Asensio et al in [4] introduced the notions of FP-gr-injective modules, then Yang and Liu in [18] investigated homological behavior of the FPgr-injective modules on gr-coherent rings. Recently in 2018, Zhao, Gao and Huang [20] gave a definition of n-presented graded modules and n-gr-coherent rings and also, by using of n-presented graded modules, they introduced the concept of n-FPgr-injective and n-gr-flat modules, and then examined the homological behavior of these modules over n-gr-coherent rings.…”

“…Recently in 2018, Zhao, Gao and Huang [20] gave a definition of n-presented graded modules and n-gr-coherent rings and also, by using of n-presented graded modules, they introduced the concept of n-FPgr-injective and n-gr-flat modules, and then examined the homological behavior of these modules over n-gr-coherent rings. In case n = 1, see [13,18]. The aim of this paper is to introduce and study n-copresented graded right modules, n-gr-cocoherent right rings and n-FCP-gr-projective right modules as a dual notion of n-presented graded left modules, n-gr-coherent left rings and n-FP-grinjective left modules, respectively.…”

Coherence relative to a weak torsion class

“…Recall that a left R-module M is called FP-injective, see [13], or absolutely pure, see [11], if Ext 1 R (A, M ) = 0 for any finitely presented left R-module A; a right R-module M is flat if and only if Tor R 1 (M, A) = 0 for any finitely presented left R-module A; a ring R is left coherent, see [1], if every finitely generated left ideal of R is finitely presented, or equivalently, if every finitely generated submodule of a projective left R-module is finitely presented. The FP-injective modules, flat modules, coherent rings and their generalizations have been studied extensively by many authors (see, for example, [1], [3], [4], [8], [10], [13], [18], [17]).…”