… relative to a class of modules C closed under submodules and factor modules. We call
such a class … For instance, if C is the class of singular modules, then we obtain the δ-supplements …

… , that a module has a flat cover if it has a flat precover and … If F and G are adjoint then we say
that F is left adjoint to G … We say that (EA,g) is an X-preenvelope of A if g ∈ Hom(A, EA) such …

… The rings whose simple right modules have a projective preenvelope are … is precovering if
and only if it is covering if and only if it is closed under … by the standard adjoint isomorphism. …

… More precisely, an n-th differential module is such an R-module with an R-linear … X
precovering in C if any M ∈ obC has an X-precover. Dually, the notions of preenvelopes …

… H⊥ to get a special H⊥preenvelope M → X0 with X0 free of … inclusion functor has a left
adjoint. We finish this section … is complete if every module has a special A-precover. Since (…

… In Theorem 9.14 we prove that fully exact subcategories of module … Notice that f is an X
-precover and g is a Y -preenvelope, so … that B is precovering since, if q is the right adjoint of the …

… The construction of the Bongartz preenvelope is neither unique … will be a left adjoint of the
embedding of P + into Mod - R . … For any module M , we fix a Bongartz preenvelope M ↪ i M 0 …

… R^{op}\) consisting of C-Gorenstein injective modules; we also get that if S is right coherent,
… ) for some flat module F in \(\mathop {\mathrm{Mod}}\nolimits S\). By the adjoint isomorphism …

… We show that this tensor product is left adjoint to the Hom … ** – L – 0 is a pure injective
preenvelope. Also we know that direct … So we only have to find Gorenstein flat precovers. Let CG …

… , then ϕ : C → M is called an F-precover of M if HomR-gr(C′,C) … Since F is a right adjoint
functor of the forgetful functor, we … every graded R-module has an FP-gr-injective preenvelope. …