Essential embedding of cyclic modules

in projectives

Authors:
José L. Gómez Pardo and Pedro A. Guil Asensio

Journal:
Trans. Amer. Math. Soc. **349** (1997), 4343-4353

MSC (1991):
Primary 16L60, 16L30; Secondary 16D50, 16E50, 16S50

DOI:
https://doi.org/10.1090/S0002-9947-97-01529-8

MathSciNet review:
1329538

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Abstract | References | Similar Articles | Additional Information

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

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Additional Information

**José L. Gómez Pardo**

Affiliation:
Departamento de Algebra, Universidad de Santiago, 15771 Santiago de Compostela, Spain

Email:
pardo@zmat.usc.es

**Pedro A. Guil Asensio**

Affiliation:
Departamento de Matematicas, Universidad de Murcia, 30100 Espinardo, Murcia, Spain

Email:
paguil@fcu.um.es

DOI:
https://doi.org/10.1090/S0002-9947-97-01529-8

Received by editor(s):
December 2, 1994

Received by editor(s) in revised form:
May 2, 1995

Additional Notes:
Work partially supported by the DGICYT (PB93-0515, Spain). The first author was also partially supported by the European Community (Contract CHRX-CT93-0091) and the Xunta de Galicia (XUGA 10502B94).

Article copyright:
© Copyright 1997
American Mathematical Society