ARE COVERING (ENVELOPING) MORPHISMS MINIMAL?

Abstract. We prove that for certain classes of modules F such that direct sums of F-covers (F-envelopes) are F-covers (F-envelopes), F-covering (F-enveloping) homomorphisms are always right (left) minimal. As a particular case we see that over noetherian rings, essential monomorphisms are left min-imal. The same type of results are given when direct products of F-covers are F-covers. Finally we prove that over commutative noetherian rings, any direct product of at covers of modules of nite length is a at cover. 1. Preliminaries This study was prompted by the following question. If f: M! M is an endomorphism of a (left) R-module such that the restriction of f to some essential submodule N is the identity on N, then is f an automorphism of M? In this paper we will show that the answer is yes if the ring R is left noetherian. In fact we will note that this question is a special case of a more general one and that our techniques provide answers in the more general situation. De nition 1.1 ([2, pgs. 6 and 8]). Let R be any associative ring (not necessaril