ON SEMIPERFECT F-INJECTIVE RINGS

Abstract

A ring R is called right F-injective if every right R-homomorphism from a finitely generated right ideal of R to R extends to an endomorphism of R. R is called a right FSE-ring if R is a right F-injective semiperfect ring with essential right socle. The class of right FSE-rings is broader than that of right PF-rings. In this paper, we study and provide some characterizations of this class of rings. We prove that if R is left perfect, right F-injective, then R is QF if and only if R/S is left finitely cogenerated where S = Sr = Sl if and only if R is left semiartinian, Soc2(R) is left finitely generated. It is also proved that R is QF if and only if R is left perfect, mininjective and J2 = r(I) for a finite subset I of R. Some known results are obtained as corollaries.