Abstract. 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 quasiFrobenius 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.

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Abstract. Let R be a ring, E = E(R R ) its injective envelope, S = End(E R ) and J the Jacobson radical of S. It is shown that if every finitely generated submodule of E embeds in a finitely presented module of projective dimension ≤ 1, then every finitley generated right S/J-module X is canonically isomorphic to Hom R (E, X ⊗ S E). This fact, together with a well-known theorem of Osofsky, allows us to prove that if, moreover, E/JE is completely pure-injective (a property that holds, for example, when the right pure global dimension of R is ≤ 1 and hence when R is a countable ring), then S is semiperfect and R R is finite-dimensional. We obtain several applications and a characterization of right hereditary right noetherian rings.

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