The Faith-Menal conjecture is one of the three main open conjectures
on QF rings. It says that every right noetherian and left FP-injective
ring is QF. In this paper, it is proved that the conjecture is true if every
nonzero complement left ideal of the ring R is not small (or not singular).
Several known results are then obtained as corollaries.
Let R be a ring. R is called a quasi-Frobenius (QF) ring if R is right artinian and RR is an
injective right R-module. In this article, we introduce (weak) fuzzy homomorphisms of
modules to obtain a fuzzy characterization of QF rings. We also obtain some fuzzy
characterizations of right artinian rings and right CF rings. These results throw new
light on the research of QF rings and the related CF conjecture.