Essential embedding of cyclic modules in projectives

Abstract

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.

ASPECTS OF WEAK, s-CS AND ALMOST INJECTIVE RINGS

“…Now suppose that J = 0. Clearly, every simple singular right R-module embeds in E. In particular, every simple quotient of E is isomorphic to a simple submodule of E, and so E is a finitely generated injective and projective module containing a copy of every simple quotient of E. By [8,Lemma 18], E has a finitely generated essential socle. Then by hypothesis, there exist simple submodules S 1 , ..., S n of E such that {S 1 , ..., S n } is a complete set of representatives of the isomorphism classes of simple singular right R-modules.…”

New characterizations of pseudo-Frobenius rings and a generalization of the FGF conjecture

“…We begin by extending in Theorem 2.1 the techniques developed in [11]. This allows us to obtain as corollaries the main results of [11,12].…”

“…Both conjectures are still open, whereas it is known that the CF conjecture implies the FGF conjecture and that they are true under many different additional hypothesis (see e.g. [9,11,13,14,17,22]). Note that every right FGF ring is a right CF ring, but the converse is not true.…”

“…This approach to the conjecture was initiated independently by Björk [5] and Tolskaya [32] who proved that every right self-injective right CF ring is right artinian. And it culminated in [11], where the authors proved that every ring in which any cyclic (resp., finitely generated) right module essentially embeds in a projective module is right artinian (resp., QF). They also proved in [12] that a right CF and right extending ring has finitely generated essential socle.…”

On the Goldie dimension of injective modules

“…The proof of our main result relies on a counting argument based on "Tarski's lemma" [15], which had been applied by Osofsky in [11] (see also [7,8,12]) to deduce the finiteness of the socle of an injective cogenerator ring. When the Goldie dimension of M is a regular infinite cardinal we show in the last part of the paper that the preceding results can be strengthened.…”