Are covering (enveloping) morphisms minimal?

We prove that for certain classes of modules F such that direct sums of F-c overs (F-envelopes) are F-covers (F-envelopes), F-covering (F-enveloping) h omomorphisms are always right (left) minimal. As a particular case we see t hat over noetherian rings, essential monomorphisms are left minimal. The sa me type of results are given when direct products of F-covers are F-covers. Finally we prove that over commutative noetherian rings, any direct produc t of at covers of modules of finite length is a at cover.