Let
R
R
be a ring and
E
=
E
(
R
R
)
E = E(R_R)
its injective envelope. We show that if every simple right
R
R
-module embeds in
R
R
R_R
and every cyclic submodule of
E
R
E_R
is essentially embeddable in a projective module, then
R
R
R_R
has finite essential socle. As a consequence, we prove that if each finitely generated right
R
R
-module is essentially embeddable in a projective module, then
R
R
is a quasi-Frobenius ring. We also obtain several other applications and, among them: a) we answer affirmatively a question of Al-Huzali, Jain, and López-Permouth, by showing that a right CEP ring (i.e., a ring
R
R
such that every cyclic right module is essentially embeddable in a projective module) is always right artinian; b) we prove that if
R
R
is right FGF (i.e., any finitely generated right
R
R
-module embeds in a free module) and right CS, then
R
R
is quasi-Frobenius.