  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 $S_{r}\subseteq ^{ess}R_{R}$;
(2) $R$ is $QF$ if and only if $R$ is a right simple injective ring with $ACC$
on right annihilators in which $S_{r}\subseteq ^{ess}R_{R}$. Several known
results on $QF$ rings are obtained as corollaries.
