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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
DOI: https://doi.org/10.1090/S0002-9947-97-01529-8
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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  • 1. A. Al-Huzali, S.K. Jain and S.R. López-Permouth, On the weak relative-injectivity of rings and modules, Lecture Notes in Math., vol. 1448, Springer-Verlag, Berlin and New York, 1989, pp. 93-98. MR 92d:16006
  • 2. J.E. Björk, Radical properties of perfect modules, J. Reine Angew. Math. 245 (1972), 78-86. MR 47:1864
  • 3. A.W. Chatters and C.R. Hajarnavis, Rings in which every complement right ideal is a direct summand, Quart. J. Math. Oxford Ser. (2) 28 (1977), 61-80. MR 55:10519
  • 4. Nguyen Viet Dung, Dinh Van Huynh, P. Smith, and R. Wisbauer, Extending modules, Pitman Res. Notes in Math., vol. 313, Longman, Harlow, 1994. MR 96f:16008
  • 5. C. Faith, Algebra. II: Ring Theory, Springer-Verlag, Berlin and New York, 1976. MR 55:383
  • 6. C. Faith, Embedding modules in projectives. A report on a problem, Lecture Notes in Math., vol. 951, Springer-Verlag, Berlin and New York, 1982, pp. 21-40. MR 84i:16001
  • 7. C. Faith, Embedding torsionless modules in projectives, Publ. Mat. 34 (1990), 379-387. MR 92b:16016
  • 8. J.L. Gómez Pardo and P.A. Guil Asensio, Endomorphism rings of completely pure-injective modules, Proc. Amer. Math. Soc. 124 (1996), 2301-2309. MR 96j:16029
  • 9. S.K. Jain and S.R. López-Permouth, A generalization of the Wedderburn-Artin theorem, Proc. Amer. Math. Soc. 106 (1989), 19-23. MR 89i:16013
  • 10. S.K. Jain and S.R. López-Permouth, Rings whose cyclics are essentially embeddable in projective modules, J. Algebra 128 (1990), 257-269. MR 90k:16016
  • 11. S.K. Jain, S.R. López-Permouth and S. Singh, On a class of QI-rings, Glasgow Math. J. 34 (1992), 75-81. MR 93e:16008
  • 12. L.S. Levy, Torsion-free and divisible modules over non-integral domains, Canad. J. Math. 15 (1963), 132-151. MR 26:155
  • 13. P. Menal, On the endomorphism ring of a free module, Publ. Mat. Univ. Autonoma Barcelona 27 (1983), 141-154. MR 86g:16046
  • 14. B.L. Osofsky, A generalization of quasi-Frobenius rings, J. Algebra 4 (1966), 373-387; errata, 9 (1968), 120. MR 34:4305; MR 36:6443

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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

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