… So a natural question is: can one deduce to the existence of (pre)covers or (pre)envelopes
of QC … At the end of this paper we show the existence of (pre)envelops and (pre)covers by In …

… In fact, we prove that AddM-covers exist for instance when AddM consists of modules …
existence of AddM-covers in case that all Mi are finitely presented. We then turn to AddM-envelopes…

… dimension of the injective envelope of any (Gorenstein) flat left R-module is at most the flat
dimension of the injective envelope of R R. Then we get that the injective envelope of R R is (…

… Our main results in Section 3 are about the existence of covers and envelopes with
respect to the class of Gorenstein injective modules. We recall the following definitions: …

… cover and a reflexive envelope of a module. We show that every finitely generated R-module
has a reflexive cover … We note that an X-cover, when it exists, is unique up to isomorphism. …

… that Ω contains all epimorphisms of K) and for any object X the existence of envelope env
Ω Ψ X is equivalent to the existence of envelope env Ω Φ X, and these envelopes coincide: …

… -envelope of M. If 3 is the class of in-jective modules, then we get the usual injective envelopes…
Hence there exists a homomorphism v from F to K such that v f = t. Since uv f = ut = f and f : …

… such that every module admits a divisible envelope or a P1-cover, where P1 is the class
of … A sufficient condition for the existence of envelopes is illustrated by the following result. …

… envelopes for any ring homomorphism R 3 S, we only have to prove that HomR TY À
preserves injective envelopes … We note that such an mt exists by 2) above. Let l be any natural …

… of their injective envelope and proved that any … envelopes and covers. In this general theory
of modules invariant under automorphisms (resp. endomorphisms) of envelopes and covers, …