WEAKLY PROJECTIVE AND WEAKLY INJECTIVE-MODULES

A module M is said to be weakly N-projective if it has a projective co ver pi:P(M) -->> M and for each homormorphism phi:P(M) --> N there exi sts an epimorphism sigma:P(M) -->> M such that phi(ker sigma) = 0, equ ivalently there exists a homomorphism phi:M --> N such that phisigma = phi. A module M is said to be weakly projective if it is weakly N-pro jective for all finitely generated modules N. Weakly N-injective and w eakly injective modules are defined dually. In this paper we study rin gs over which every weakly injective right R-module is weakly projecti ve. We also study those rings over which every weakly projective right module is weakly injective. Among other results, we show that for a r ing 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-ri ng such that for any indecomposable projective right module eR and for any right ideal 1, soc(eR/eI) = (eR/eJ)n for some positive integer n. (4) R is right artinian ring and every weakly injective right R-modul e is weakly projective. (5) Every weakly projective right R-module is weakly injective and every weakly injective right R-module is weakly p rojective.