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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
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.

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  • 1 Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, Springer-Verlag, New York-Heidelberg, 1974. Graduate Texts in Mathematics, Vol. 13. MR 0417223
  • 2 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), no. 109, 61–80. MR 0437595
  • 3 John Clark and Dinh van Huynh, When is a self-injective semiperfect ring quasi-Frobenius?, J. Algebra 165 (1994), no. 3, 531–542. MR 1275918, 10.1006/jabr.1994.1128
  • 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 the splitting properties, American Mathematical Society, Providence, R. I., 1972. Memoirs of the American Mathematical Society, No. 124. MR 0340335
  • 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 Friedrich Kasch, Moduln und Ringe, B. G. Teubner, Stuttgart, 1977. Mathematische Leitfäden. MR 0429963
  • 10 Saad H. Mohamed and Bruno J. Müller, Continuous and discrete modules, London Mathematical Society Lecture Note Series, vol. 147, Cambridge University Press, Cambridge, 1990. MR 1084376
  • 11 Kiyoichi Oshiro, Lifting modules, extending modules and their applications to QF-rings, Hokkaido Math. J. 13 (1984), no. 3, 310–338. MR 764267, 10.14492/hokmj/1381757705
  • 12 B. L. Osofsky, A generalization of quasi-Frobenius rings, J. Algebra 4 (1966), 373–387. MR 0204463

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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

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