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A note on quasi-Frobenius rings


Authors: Dinh Van Huynh and Ngo Si Tung
Journal: Proc. Amer. Math. Soc. 124 (1996), 371-375
MSC (1991): Primary 16L60, 16D50
DOI: https://doi.org/10.1090/S0002-9939-96-03303-5
MathSciNet review: 1322929
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Abstract: It is shown that a semiperfect ring $R$ is quasi-Frobenius if and only if every closed submodule of $R(\omega )$ is non-small, where $R(\omega )$ denotes the direct sum of $\omega $ copies of the right $R$-module $R$ and $\omega $ is the first infinite ordinal.


References [Enhancements On Off] (What's this?)

  • 1 F.W. Anderson and K.R. Fuller, Rings and Categories of Modules, Springer-Verlag, Berlin-New York, 1974, MR 54:5281.
  • 2 A.W. Chatters and C.R. Hajarnavis, Rings in which every complement right ideal is a direct summand, Quart. J. Math. Oxford 28 (1977), 61--80, MR 55:10519.
  • 3 J. Clark and D.V. Huynh, When is a self-injective semiperfect ring quasi-Frobenius?, J. Algebra 164 (1994), 531--542, MR 95d:16006.
  • 4 C. Faith, Algebra II: Ring Theory, Springer-Verlag, Berlin - New York, 1976.
  • 5 C. Faith, When self-injective rings are QF: A report on a problem, Centre Recerca Matematica Institut d'Estudis Catalans, (Spain), 1990.
  • 6 K.R. Goodearl, Singular torsion and splitting properties, Mem. Amer. Math. Soc. 124 (1972), MR 49:5090.
  • 7 M. Harada, Non-small modules and non-cosmall modules, Proc. of the 1978 Antw. Conf. Mercel Dekker, pp. (669--689).
  • 8 D.V. Huynh, A right countably sigma-CS ring with ACC or DCC on projective principal right ideals is left artinian and QF-3, Trans. Amer. Math. Soc. (to appear).
  • 9 F. Kasch, Moduln und Ringe, Teubner, Stutgart, 1977, MR 55:2971.
  • 10 S.H. Mohamed and B.J. Müller, Continuous and Discrete Modules, London Math. Soc. Lecture Note Series 147, Cambridge Univ. Press, 1990, MR 92b:16009.
  • 11 K. Oshiro, Lifting modules, extending modules and their applications to QF-rings, Hokkaido Math. J. 13 (1984), 310--338, MR 86b:16008a.
  • 12 B.L. Osofsky, A generalization of quasi-Frobenius rings, J. Algebra 4 (1966), 373--387, MR 34:4305.

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Additional Information

Dinh Van Huynh
Affiliation: Institute of Mathematics, P. O. Box 631 Boho, Hanoi, Vietnam

Ngo Si Tung
Affiliation: Institute of Mathematics, P. O. Box 631 Boho, Hanoi, Vietnam
Email: huynh@math.ohio-state.edu

DOI: https://doi.org/10.1090/S0002-9939-96-03303-5
Keywords: Closed submodules, small modules, non-small modules, quasi-\linebreak Frobenius rings
Received by editor(s): August 22, 1994
Communicated by: Ken Goodearl
Article copyright: © Copyright 1996 American Mathematical Society

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