… Have similar result for preenvelopes. … Theorem Every group H admits an J-precover. If
J includes all epimorphisms g : F→→F with F ≤ F, and F ∈ J, then each group admits an J-cover. …

… the proper context for the study of precovers and preenvelopes (approximation theory).
Expositions of approximation theory for categories of modules may be found in the monographs …

… find a projective module P and an epimorphism P → f*(X). Thus, by the adjoint situation we
… Precovers, covers, preenvelopes and envelopes are often named according to the class ℱ. …

… constructed by precovers and preenvelopes, respectively. … C, we use AddR(C) to denote
the class of all R-modules which … We give in this section some transfer results under adjoint …

… class of R-modules, by an F-preenvelope of an R-module M we … Dually we have the definitions
of an F-precover and an F-cover… SCR with R = EndS C we have the adjoint pair of functors …

… modules, the modules that occur as zero cohomologies of 2-term silting complexes. Silting
modules … than just the existence of preenvelopes and precovers, because one always has …

… Finally, in Section 5, we examine the K ( PInj ) -preenvelopes … -precover, we say that the
subcategory D ⊆ C is precovering… of a right adjoint to the embedding K ( PProj ) ↪ K ( R ) . □ …

… modules occurring in this category are the Prüfer modules, … An S-preenvelope f : M → X of M
is an S-envelope if whenever gf … An S-precover f : M → X of M is an S-cover if whenever fg = f …

… left R-modules. For the categories of right R-modules and complexes of right R-modules, we
… The class F is called precovering, covering or special precovering if every M ∈ Mod(R) has …

… preenvelopes. By constructing duality pairs with respect to Gorenstein (L, A)-injective
modules and Gorenstein (L, A)-flat modules … X is precovering if every object of A has an X-precover. …