Ring extensions, injective covers and envelopes

The present paper is devoted to the study of those rings Ii such that for a ny ring homomorphism R-S the functor Hom(R)(S,-) : R-Mod--> S-Mod preserves injective envelopes or injective covers. The case of injective envelopes has been studied by T. Wurfel ([9]). who ga ve a characterization of such rings (Theorem 10). In this paper we give ano ther characterization of those rings in Section 2. One of the tools we use is a generalization of a certain type of module initially studied by Northc ott ([4]), McKerrow ([3]) and Park ([5] and [6]). The case of injective covers is treated in Section 3. where we give a compl ete characterization of commutative noetherian rings satisfying the propert y mentioned above.