Semi-Primary QF-3 Rings

Source 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].