A note on quasi-Frobenius rings

Huynh, Dinh; Tung, Ngo

doi:10.1090/S0002-9939-96-03303-5

A note on quasi-Frobenius rings
HTML articles powered by AMS MathViewer

by Dinh Van Huynh and Ngo Si Tung

Proc. Amer. Math. Soc. 124 (1996), 371-375

DOI: https://doi.org/10.1090/S0002-9939-96-03303-5

PDF | 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