We prove that (a) if R is left coherent, then a finitely presented right R–module M is copure
flat if and only if M is a cokernel of a flat preen-velope (b) if R is a ring and na nonnegative …

Let A be a noetherian Auslander regular ring and δ the canonical dimension function on A-
modules, which is defined as δ (M)= d− j (M) where d is the global dimension of A and j (M) …

Let R be a ring, na fixed nonnegative integer and In (Fn) the class of all left (right) R-modules
of injective (flat) dimension at most n. A left R-module M (resp., right R-module F) is called n …

For a ring R, we prove that all left R-modules have weak injective covers and weak injective
preenvelopes. Then we characterize the right weak injective dimension of modules in terms …

We characterize left Noetherian rings in terms of the duality property of injective
preenvelopes and flat precovers. For a left and right Noetherian ring R, we prove that the flat …

In this paper, we study the existence of L⊥-envelopes, L-envelopes, D⊥-envelopes, D-
covers, and L-covers where L and D denote the classes of modules of injective and …

Bass characterized the rings R with the property that every left R-module has a projective
cover. These are the left perfect rings. A ring is left perfect if and only if the class of projective …

In this paper, we show the existence of copure injective preenvelopes over noetherian rings
and copure flat preenvelopes over commutative artinian rings. We use this to characterize …

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) …

A module M is said to be weakly injective if every finitely generated submodule of its
injective hull E (M) is contained in a submodule X of E (M) isomorphic to M. We prove that a …