Introduction* The concept of the" injective envelope" of a module was first given by Eckmann and Schopf [2], although this terminology was first employed by Matlis [5]. The injective envelope of a module always exists and is unique in a certain sense. The dual concept of the" projective cover" of a module has been given by Bass [1], and the concept of" minimal epimorphism" in a" perfect category" as defined by Eilenberg [3] is a particular case of this concept. The projective cover of a module does not always exist but is unique whenever it exists. Eilenberg [3] has proved that every module in a perfect category possesses a projective cover. Bass [1] calls a ring" perfect" if every module over the ring possesses a projective cover, and he gives several characterizations of a perfect ring. We shall call a module" perfect" if it possesses a projective cover. It would be natural to try to characterize a perfect module, but it seems likely that such attempts may result in obtaining equivalent definitions of the projective cover of a module. One might instead consider specific types of modules and try to obtain necessary and sufficient conditions so that they may be perfect. In § 1 we first define a category of perfect modules and then give a necessary and sufficient condition for a finitely generated module over a Noetherian ring to be perfect. In § 2 we give some results on" essential monomorphism" and" minimal epimorphism"[1, 2]. In § 3 we give some results on modules over perfect rings. In § 4 we give new proofs of some known results to show how the concepts of the injective envelope and the projective cover of a module simplify the proofs considerably. I should like to thank Professor Cartan under whose guidance this work was done. I should also like to thank Pierre Gabriel with whom I have had interesting discussions on the subject. l A category of perfect modules. Let A be a ring with unit element 1^ 0. Throughout this paper we shall be concerned with unitary left A-modules and so we shall call them simply modules. We recall some definitions. Let f: L—> M be a homomorphism of modules. If JTΠ iw/= 0 implies H= 0, where H is a submodule of M, f is called an essential homomorphism; moreover, if/is a monomorphism and M is an injective module, then M is called the injective envelope of L and is denoted by E (L). If, however, K+ Ker f—L implies K= L, where