Essential embedding of cyclic modules in projectives

Abstract

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.