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Quasi-Frobenius X-Rings

Published online by Cambridge University Press:  20 November 2018

Abraham Zaks
Abraham Zaks*
Affiliation:
Technion I.I.T., Haifa, Israel
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In a recent study of a specific class of quasi-Frobenius rings, Feller has found it useful to introduce the X-rings ([3]). He suggested among others the following topics:

  1. (A) Determine the properties of completely indecomposable rings and matrix rings over completely indecomposable rings.

  2. (B) Determine the properties of modules over quasi-Frobenius X-rings.

We point out that the completely indecomposable rings are the local quasi-Frobenius rings. Problems (A) and (B) then lead naturally to semi-local quasi-Frobenius rings, and to matrix algebra over local quasi-Frobenius rings. These types of rings are discussed in sections 1 and 2.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 1 , March 1970 , pp. 23 - 30
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Faith, C., Rings with ascending condition on annihilators, Nagoya Math. J. 27 (1966), 179-191.Google Scholar
2. Faith, C., and Walker, E. A., Direct sum representation ofinjective modules, J. Alg. 5 (1967), 203-221.Google Scholar
3. Feller, E. H., A type of quasi-Frobenius ring, Canad. Math. Bull. 10 (1967), 19-27.Google Scholar
4. Kaplansky, I., Projective modules, Ann. Math. 68 (1958), 372-377.Google Scholar
5. Morita, K., Duality for modules and its applications to the theory of rings with minimum conditions, Sci. Rep., Tokyo Kyoiku Daigaku 6, No. 150 (1958), 83-142.Google Scholar
6. Pollingher, A. and Zaks, A., Some remarks on quasi-Frobenius rings, J. Alg. 10 (1968), 231-240.Google Scholar
7. Rosenberg, A. and Zelinsky, D., Annihilators, Portugaliae Mathematica 20, Fasc. 1 (1961), 53-65.Google Scholar
8. Zaks, A., Restricted Gorenstein rings, Ill. J. Math., 13 (1969), 796-803.Google Scholar
9. Zaks, A., Isobaric QF-rings, to be published.Google Scholar
10. Lambek, J., Lectures on rings and modules, Blaisdell Publishing Company, Waltham. Massachusetts, 1966.Google Scholar