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A Characterization of QF-3 Rings

Published online by Cambridge University Press:  22 January 2016

L. E. T. WU ,
H. Y. Mochizuki  and
J. P. Jans
L. E. T. WU
Affiliation:
Western Washington State College
H. Y. Mochizuki
Affiliation:
University of California
J. P. Jans
Affiliation:
University of Washington
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A left QF-3 ring R is one in which RR, the ring considered as a left module over itself, can be embedded in a projective infective left R module Q(RR). QF-3 rings were introduced by Thrall [14] and have been studied and characterized by a number of authors [5, 8, 9, 12, 13, 15] usually restricted to the case of algebras over a field. In such a case, the concept of left QF-3 and right QF-3 coincide.

The study of QF-3 rings and algebras and many other such classes of rings had its origin in the now classic papers of Nakayama [10, 11]. He was an outstanding pioneer in algebra for many years, and we acknowledge our great debt to him and to his many excellent papers.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 27 , Issue 1 , February 1966 , pp. 7 - 13
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1966

References

[1] Bass, H., Finitistic dimension and a homological generalization of semi-primary rings, Trans. Am. Math. Soc, 95 (1960), pp. 446–488.Google Scholar
[2] Chase, S. U., Direct products of modules, Trans. Am. Math. Soc, 97 (1960), pp. 457–473.CrossRefGoogle Scholar
[3] Dickson, S. E., A torsion theory for Abelian categories, (to appear).Google Scholar
[4] Eckmann, B. and Schopf, A., Über injective Moduln, Archiv der Math., 4 (1953), pp. 75–78.Google Scholar
[5] Jans, J. P., Projective injective modules, Pacific Journal of Math., 9 (1959), pp. 1103–1108.Google Scholar
[6] Jans, J. P., Rings and homology, Holt, Rinehart and Winston, New York, (1964).Google Scholar
[7] Mochizuki, H. Y., Finistic Global Dimension for Rings, Pacific J. Math. 15 (1965), pp. 249–258.Google Scholar
[8] Mochizuki, H. Y., On the double commutator algebra of QF-3 algebras (to appear).Google Scholar
[9] Morita, K., Duality for modules and its applications to the theory of rings with minimum condition, Sci. Rep. Tokyo Kyoiku Daigaku, 6 (1958), No. 150, pp 83–142.Google Scholar
[10] Nakayama, T., On Frobeniusean Algebras I, Annals of Math., 40 (1939), pp. 611–633.Google Scholar
[11] Nakayama, T., On Frobeniusean Algebras II, Annals of Math., 42 (1941), pp. 1–21.Google Scholar
[12] Tachikawa, H., A characterization of QF-3 algebras, Proc. Am. Math. Soc, 13 (1962), pp. 101–103.Google Scholar
[13] Tachikawa, H., On the dominant dimensions of QF-3 algebras, Trans. Am. Math. Soc. 112 (1964), pp. 249–266.Google Scholar
[14] Thrall, R. M., Some generalizations of Quasi-Frobenius algebras, Trans. Am. Math, Soc, 64 (1948), pp. 173–183,Google Scholar
[15] Wall, D. W., Algebras with unique minimal faithful representations, Duke Math. J., 25 (1958), pp. 321–329.CrossRefGoogle Scholar
[16] Wu, L. E. T., A characterization of self injective rings, (to appear in the Illinois J. of Math.).Google Scholar