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Semi-Primary QF-3 Rings

Published online by Cambridge University Press:  22 January 2016

R. R. Colby  and
Edgar A. Rutter Jr.
R. R. Colby
Affiliation:
The University of Kansas
Edgar A. Rutter Jr.
Affiliation:
The University of Kansas
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A ring R (with identity) is semi-primary if it contains a nilpotent ideal N with R/N semi-simple with minimum condition. R is called a left QF-3 ring if it contains a faithful projective injective left ideal. If R is semi-primary and left QF-3, then there is a faithful projective injective left ideal of R which is a direct summand of every faithful left R-module [5], in agreement with the definition of QF-3 algebra given by R.M. Thrall [6]. Let Q(M) denote the injective envelope of a (left) R-module M. We call R left QF-3+ if Q(R) is projective. J.P. Jans showed that among rings with minimum condition on left ideals, the classes of QF-3 and QF-3+ rings coincide [5].

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 32 , June 1968 , pp. 253 - 258
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1968

References

[1] Bass, H.: Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960) 466–488.CrossRefGoogle Scholar
[2] Eilenberg, S.: Homological dimension and syzygies, Ann. of Math. 64 (1956) 328–336.CrossRefGoogle Scholar
[3] Goldie, A.W.: Torsion-free modules and rings, J.A. 1 (1964) 268–287.Google Scholar
[4] Harada, M.: QF-3 and semi-primary PP rings II, Osaka J. Math. 3 (1966) 21–27.Google Scholar
[5] Jans, J.P.: Projective injective modules, Pac. J. Math. 9 (1959), 1103–1108.CrossRefGoogle Scholar
[6] Thrall, R.M.: Some generalizations of quasi-Frobenius algebras, Trans. Amer. Math. Soc. 64 (1948) 173–183.Google Scholar
[7] Wu, L.E.T., Mochizuki, H.Y., and Jans, J.P.: A characterization of QF-3 rings, Nagoya Math. J. 27 (1966) 7–13.Google Scholar