A note on quasi-Frobenius rings
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- by Dinh Van Huynh and Ngo Si Tung PDF
- Proc. Amer. Math. Soc. 124 (1996), 371-375 Request permission
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
- Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, Graduate Texts in Mathematics, Vol. 13, Springer-Verlag, New York-Heidelberg, 1974. MR 0417223, DOI 10.1007/978-1-4684-9913-1
- 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 437595, DOI 10.1093/qmath/28.1.61
- 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, DOI 10.1006/jabr.1994.1128
- C. Faith, Algebra II: Ring Theory, Springer-Verlag, Berlin - New York, 1976.
- C. Faith, When self-injective rings are QF: A report on a problem, Centre Recerca Matematica Institut d’Estudis Catalans, (Spain), 1990.
- K. R. Goodearl, Singular torsion and the splitting properties, Memoirs of the American Mathematical Society, No. 124, American Mathematical Society, Providence, R.I., 1972. MR 0340335
- M. Harada, Non-small modules and non-cosmall modules, Proc. of the 1978 Antw. Conf. Mercel Dekker, pp. (669–689).
- 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).
- Friedrich Kasch, Moduln und Ringe, Mathematische Leitfäden, B. G. Teubner, Stuttgart, 1977. MR 0429963, DOI 10.1007/978-3-663-05703-1
- 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, DOI 10.1017/CBO9780511600692
- Kiyoichi Oshiro, Lifting modules, extending modules and their applications to QF-rings, Hokkaido Math. J. 13 (1984), no. 3, 310–338. MR 764267, DOI 10.14492/hokmj/1381757705
- B. L. Osofsky, A generalization of quasi-Frobenius rings, J. Algebra 4 (1966), 373–387. MR 204463, DOI 10.1016/0021-8693(66)90028-7
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
- Received by editor(s): August 22, 1994
- Communicated by: Ken Goodearl
- © Copyright 1996 American Mathematical Society
- 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