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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
MathSciNet review: 1329538
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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 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.


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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: http://dx.doi.org/10.1090/S0002-9947-97-01529-8
PII: S 0002-9947(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