Weakly Projective and Weakly Injective Modules

Abstract A module M is said to be weakly N-projective if it has a projective cover π: P(M) ↠M and for each homomorphism : P(M) → N there exists an epimorphism σ:P(M) ↠M such that (kerσ) = 0, equivalently there exists a homomorphism :M ↠N such that σ= . A module M is said to be weakly projective if it is weakly N-projective for all finitely generated modules N. Weakly N-injective and weakly injective modules are defined dually. In this paper we study rings over which every weakly injective right R-module is weakly projective. We also study those rings over which every weakly projective right module is weakly injective. Among other results, we show that for a ring R the following conditions are equivalent: (1) R is a left perfect and every weakly projective right R-module is weakly injective. (2) R is a direct sum of matrix rings over local QF-rings. (3) R is a QF-ring such that for any indecomposable projective right module eR and for any right ideal I, soc(eR/eI) = (eR/eJ)n for some positive integer n. (4) R is right artinian ring and every weakly injective right R-module is weakly projective. (5) Every weakly projective right R-module is weakly injective and every weakly injective right R-module is weakly projective.