… Note that if f : X −→ M is an Xprecover (or an X-cover) of M, … Similarly, we use the notion of
preenvelope and envelope to … Note also that we have another adjoint pair (Ae⊗eAe −, HomA(…

… modules in the category of all modules in terms of the feasibility of rugged precovers and
covers for arbitrary modules… M→M⊕P is an injective rugged preenvelope. Furthermore, M has a …

… There exist several important pairs of adjoint functors … M-module and Gpd(X) < ∞.
Section 4 is devoted to the connections between preenvelopes (respectively, precovers) in the …

… flat modules as well as its left adjoint f∗, so f∗ preserves flat precovers by Proposition 2.10
applied to the adjoint … We observe that f∗ preserves cotorsion preenvelopes under the same …

… Relative homological algebra uses general precovering/preenveloping classes in place of
… module T can be obtained from an iteration of special T -preenvelopes of the regular module …

… The C-precover f is called a C-cover of M if any endomorphism α: C → C with fα = f is an
isomorphism. The concepts of a C-preenvelope … Hence via the Adjoint Isomorphism, we have the …

… Relative homological algebra uses general precovering/preenveloping classes in place of
… module T can be obtained from an iteration of special T -preenvelopes of the regular module …

… module occurs like that in an essentially unique way (Theorem 6.2, Theorem 6.8). In order to
do so, we employ a model structure associated with the cotilting module (… cotilting module C …

… modules consists of all short exact sequences of R-modules which we denote by AbsR-Mod.
For a proper class P of R-modules, call a submodule A of a module … injective preenvelope of …

… adjoints will lead to the existence of Gorenstein injective preenvelopes/Gorenstein projective
precovers… It is known that, this functor admits both left and right adjoints, which we will …