Harada calls a ring
R
R
right simple-injective if every
R
R
-homomorphism with simple image from a right ideal of
R
R
to
R
R
is given by left multiplication by an element of
R
R
. In this paper we show that every left perfect, left and right simple-injective ring is quasi-Frobenius, extending a well known result of Osofsky on self-injective rings. It is also shown that if
R
R
is left perfect and right simple-injective, then
R
R
is quasi-Frobenius if and only if the second socle of
R
R
is countably generated as a left
R
R
-module, extending many recent results on self-injective rings. Examples are given to show that our results are non-trivial extensions of those on self-injective rings.