A Note on Quasi-Frobenius Rings

The Faith-Menal conjecture says that every strongly right Johns ring is QF . The conjecture is also equivalent to say every right noetherian left FP -injective ring is QF . In this short article, we show that the conjecture is true under the condition( a proper generalization of left CS condition) that every nonzero complement left ideal is not small( a left ideal I is called small if for every left ideal K, K+I=R implies K=R). It is also proved that (1) R is QF if and only if R is a left and right mininjective ring with ACC on right annihilators in which Sr ⊆ ess RR; (2) R is QF if and only if R is a right simple injective ring with ACC on right annihilators in which Sr ⊆ ess RR. Several known results on QF rings are obtained as corollaries.