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The Structure of Quasi-Frobenius Rings

Published online by Cambridge University Press:  20 November 2018

Bruno J. Müller
Bruno J. Müller*
Affiliation:
McMaster University, Hamilton, Ontario
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Utilizing a matrix representation of semiperfect rings by a family of bimodules over local rings, we describe the structure of generalized quasi-Frobenius rings in two steps: a cyclic generalized quasi-Frobenius ring is a matrix ring over a cycle of Morita dualities between local rings, and an arbitrary generalized quasi-Frobenius ring is a matrix ring over a family of cyclic generalized quasi-Frobenius rings.

Our results provide a complete classification of generalized quasi-Frobenius rings, modulo the classification of local rings with Morita duality, of certain bimodules over such rings, and of certain rest families of multiplication maps.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 5 , October 1974 , pp. 1141 - 1151
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Azumaya, G., Completely faithful modules and self-infective rings, Nagoya Math. J. 27 (1966), 697–708.Google Scholar
2. Eilenberg, S. and T. Nakayama, On the dimension of modules and algebras, II, Nagoya Math. J. 9 (1955), 1–16.Google Scholar
3. Fried, E., Beitrage zur Théorie der Frobenius-Algebren, Math. Ann. 155 (1964), 265–269.Google Scholar
4. Fuller, K., On indecomposable infectives over Artinian rings, Pacific J. Math. 29 (1969), 115–135.Google Scholar
5. Hannula, A. T., On the construction of quasi-Frobenius rings, J. Algebra 25 (1973), 403–414.Google Scholar
6. Hannula, A. T., The Morita context and the construction of QF rings, Proc. Ohio State Conf. on Orders, Group Rings and Related Topics, Lecture Note in Math. 353, Springer, (1973), 113–130.Google Scholar
7. Müller, B. J., On semiperfect rings, Illinois J. Math. 14 (1970), 464–467.Google Scholar
8. Müller, B. J., Linear compactness and Morita duality, J. Algebra 16 (1970), 60–66.Google Scholar
9. Nakayama, T., On Frobeniusean algebras. I, II, Ann. Math. 40 (1939), 611–633; 42 (1941), 1–21.Google Scholar
10. Osofsky, B. L., A generalization of quasi-Frobenius rings, J. Algebra 4 (1966), 373–387; 9 (1968), 120.Google Scholar
11. Roux, B., Sur la dualité de Morita, Tôhoku Math. J. 23 (1971), 457–472.Google Scholar
12. Utumi, Y., Self-injective rings, J. Algebra 6 (1967), 56–64.Google Scholar