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QF - 1 Rings of Global Dimension ≦ 2

Published online by Cambridge University Press:  20 November 2018

Claus Michael Ringel
Claus Michael Ringel*
Affiliation:
Mathematisches Institut der Universität, Tubingen, Germany; Carleton University, Ottawa, Ontario
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R. M. Thrall [10] introduced QF — 1, QF — 2 and QF — 3 rings as generalizations of quasi-Frobenius rings. (For definitions, see section 1. It should be noted that all rings considered are assumed to be left and right artinian.) He proved that QF — 2 rings are QF — 3 and asked whether all QF — 1 rings are QF — 2, or, at least, QF — 3. In [9] we have shown that QF — 1 rings are very similar to QF — 3 rings. On the other hand, K. Morita [6] gave two examples of QF — 1 rings, one of them not QF — 2 and therefore not QF — 3, the other one QF — 3, but not QF — 2.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 25 , Issue 2 , April 1973 , pp. 345 - 352
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Auslander, M., Representation dimension of Artin algebras (Queen Mary College Math. Notes, London).Google Scholar
2. Fuller, K. R., The structure of QF - 3 rings, Trans. Amer. Math. Soc. 134 (1968), 343–354.Google Scholar
3. Fuller, K. R., Generalized uniserial rings and their Kupisch series, Math. Z. 106 (1968), 248-260.Google Scholar
4. Fuller, K. R., Double centralizers of infectives and profectives over artinian rings, Illinois J. Math. 14 (1970), 658–664.Google Scholar
5. Jans, J. P., Projective infective modules, Pacific J. Math. 9 (1959), 1103–1108.Google Scholar
6. Morita, K., On algebras for which every faithful representation is its own second commutator, Math. Z. 69 (1958), 429–434.Google Scholar
7. Morita, K. and Tachikawa, H., QF — 3 rings (to appear).Google Scholar
8. Nakayama, T., On algebras with complete homology, Abh. Math. Sem. Univ. Hamburg 22 (1958), 300–307.Google Scholar
9. Ringel, C. M., Socle conditions for QF — 1 rings (to appear in Pacific J. Math.).Google Scholar
10. Thrall, R. M., Some generalizations of quasi-Frobenius algebras, Trans. Amer. Math. Soc. 64 (1948), 173–183.Google Scholar