Essential embedding of cyclic modules in projectives

Let $R$ be a ring and $E = E(R_R)$ its injective envelope. We show that if every simple right $R$-module embeds in $R_R$ and every cyclic submodule of $E_R$ is essentially embeddable in a projective module, then $R_R$ has finite essential socle. As a consequence, we prove that if each finitely generated right $R$-module is essentially embeddable in a projective module, then $R$ 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 $R$ such that every cyclic right module is essentially embeddable in a projective module) is always right artinian; b) we prove that if $R$ is right FGF (i.e., any finitely generated right $R$-module embeds in a free module) and right CS, then $R$ is quasi-Frobenius.