… the category of all special C-preenvelopes of M endowed with the natural subobject order
… , B), each module has a special A-precover iff each module has a special B-preenvelope; in …

… A cotorsion pair (A, B) is said to be complete if any R-module has a special B-preenvelope,
or equivalently, any Rmodule has a special A-precover. A cotorsion pair (A, B) is said to be …

… projective precover. The existence of Gorenstein preenvelopes and precovers ensures that
… C, which is a generalization of the relative tilting modules given by Auslander and Solberg. …

… ), then it is called an F-precover of M. Envelopes and preenvelopes in G are defined dually
(… , (- ⊗ F,HomA(F,-)) is an adjoint pair. This causes that we have a good notion of purity (known …

… precovers to Gorenstein injective preenvelopes. It thus follows from the above remarks that
G(M) is also an Artinian module. … where we are using the usual adjoint isomorphism. But syzd(…

… called precovering if every M ∈ G has an Aprecover. Dualizing the definition, we obtain the
notion of an A-preenvelope: … However, this functor, being right adjoint to an exact functor (the …

… classes in the category of finitely generated modules on a τ−tilting infinite … radical and is
right adjoint to the inclusion functor T →A. … Preenvelopes and precovers are also known as left …

… i(−) is left adjoint to t(−) … preenvelopes and precovers implies the existence of envelopes
and covers, and therefore add M is functorially finite, as long as M is a finite dimensional module. …

… Becker, generalizing it from module categories to some locally … The cotorsion pair ( L , R ) is
said to admit special preenvelope … the special precover and special preenvelope sequences. …

… Dually are defined X-preenvelopes and X-… of modules G such that C= G⊥). This is due to
the fact that a cotorsion pair generated by a set always provide for precovers and preenvelopes…