Let m and n be fixed positive integers and M a right R-module. Recall that M is said to be (m,
n)-injective if for any (m, n)-presented right R-module P; M is called (m, n)-flat if for any (m, n) …

Let N be a left R-module with the endomorphism ring S= End (RN). Given two cardinal
numbers α and β and a matrix A∈ S β× α, N is called flat relative to A in case, for each x∈ lN …

Let R be a ring, n, d be fixed non-negative integers, the class of (n, d)-injective left R-
modules, and the class of (n, d)-flat right R-modules. In this paper, we prove that if R is a left …

Let $ n, d $ be two non-negative integers. A left $ R $-module $ M $ is called $(n, d) $-
injective, if ${\rm Ext}^{d+ 1}(N, M)= 0$ for every $ n $-presented left $ R $-module $ N $. A …

In this paper, we investigate the notions of X⊥‐projective, X‐injective, and X‐flat modules
and give some characterizations of these modules, where X is a class of left modules. We …

In this paper, we define a class of relative derived functors in terms of left or right weak flat
resolutions to compute the weak flat dimension of modules. Moreover, we investigate two …

A submodule N of a module M is called ${\mathcal {S}} $-closed (in M) if M/N is nonsingular.
It is well-known that the class Closed of short exact sequences determined by closed …

Let R be a ring. A right R-module A is said to be C-flat if the kernel of any epimorphism B→ A
is C-pure in B, ie the induced map Hom (C, B)→ Hom (C, A) is surjective for any cyclic right …