Semi-Primary QF-3 Rings

Abstract

A ring R (with identity) is semi-primary if it contains a nilpotent ideal N with R/N semi-simple with minimum condition. R is called a left QF-3 ring if it contains a faithful projective injective left ideal. If R is semi-primary and left QF-3, then there is a faithful projective injective left ideal of R which is a direct summand of every faithful left R-module [5], in agreement with the definition of QF-3 algebra given by R.M. Thrall [6]. Let Q(M) denote the injective envelope of a (left) R-module M. We call R left QF-3+ if Q(R) is projective. J.P. Jans showed that among rings with minimum condition on left ideals, the classes of QF-3 and QF-3+ rings coincide [5].