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Abstract: Using the dual of a categorical definition of an injective envelope, injective covers can be defined. For a ringR, every leftR-module is shown to have an injective cover if and only ifR is left noetherian. Flat envelopes are defined and shown to exist for all modules over a regular local ring of dimension 2. Using injective covers, minimal injective resolvents can be defined.

Abstract: 0. Throughout this paper we assume that every ring has a unit element. A module is called injective if it is a direct summand of every extension module. A ring is said to be left self injective if it is iniective as the left module over itself. The main results we shall show in the present paper are the following: Let S be a left self injective ring. Then S/N(S) is also left self injective, where N(S) denotes the Jacobson radical of S. Any system of orthogonal idempotents of S/N(S) can be lifted to a system of orthogonal idempotents of S. This theorem about orthogonal idempotents can be proved under a somewhat weaker assumption than the left self injectivity of S. In fact, it is enough to suppose that S is a ring satisfying the following two conditions: 0.1. CONDITION. For any left ideal A there is an idempotent e such that Se is an essential extension of A.