… object and G is the category of modules over a ring R with 1 … A has a flat precover and a
cotorsion preenvelope. In Section … This is an exact functor with an exact right adjoint Sc : G −→ A …

… modules. In these notes, we give a brief introduction to the fairly recent concepts of silting
and cosilting module, … We will consider silting modules, that is, the modules that arise as zero …

… precovers and preenvelopes we define Gorenstein injective and projective dimensions in
A#H-Mod and Ah -Mod in terms of resolutions by these modules… If we consider the adjoint pair (…

… precovering, covering or special precovering if every module has an A-precover, an A-cover
or a special A-precover… G has a special A-precover and a special B-preenvelope. Similarly,(A…

… C-precovers or right C-approximations, and of C-preenvelopes … if every module has a
Gorenstein-projective precover. The … inclusion functor B→A having a right adjoint [11, Theorem …

… However, there are also substantial differences here—for instance special precovers and
preenvelopes provided by cotorsion pairs are, unlike adjoint functors coming from localizations…

… is closed under direct limits, the existence of flat precovers is … if and only if every right A-module
has a flat preenvelope. … In case of a pair of adjoint functors, we have the following …

… Fλ for the left adjoint and Fρ for the right adjoint, when they exist. … that every module has an
X-preenvelope and let M be a module. … If every module has an X-precover then we can take …

… 5.3], the class of C-projective modules is precovering. So M has a P C -precover, f : Q → M
with Q ∈ P … Let M be the cokernel of the P C n -preenvelope f : K → Q , so there exists an exact …

… of Δ -modules using (complete) duality pairs of A -modules and B -… modules relative to
complete duality pairs of Δ -modules. Finally, we give an application to Ding projective modules. …