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The Centers of Semi-Simple Algebras over a Commutative Ring

Published online by Cambridge University Press:  22 January 2016

Shizuo Endo  and
Yutaka Watanabe
Shizuo Endo
Affiliation:
Tokyo University of Education
Yutaka Watanabe
Affiliation:
Tokyo University of Education
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Recently A. Hattori introduced in [7], [8], [9] the notion of simple algebras over a commutative ring. Especially, in [9], he examined, as a fundamental problem on simple algebras, whether a directly indecomposable semi-simple algebra is simple or not and gave affirmative answers to this in some particular cases. In this note we shall first prove, as a complete answer to this, that any directly indecomposable semi-simple algebra over a Noetherian ring is simple.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 30 , August 1967 , pp. 285 - 293
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1966

References

[1] Auslander, M. and Goldman, O., Maximal orders, Trans. Amer. Math. Soc, 97 (1960), 1–24.CrossRefGoogle Scholar
[2] Auslander, M. and Goldman, O., The Brauer group of a commutative ring, Trans. Amer. Math. Soc, 97 (1960), 367–409.Google Scholar
[3] Bass, H., Projective modules over algebras, Ann. of Math., 73 (1963), 532–542.Google Scholar
[4] Cohn, P.M., A remark on matrix rings over free ideal rings, Proc. Camb. Phil. Soc, 62 (1966), 1–4.Google Scholar
[5] Endo, S., Completely faithful modules and quasi-Frobenius algebras, To appear.Google Scholar
[6] Gabriel, P., Des catégories abéliennes, Bull. Soc. Math. France, 90 (1962), 323–428.Google Scholar
[7] Hattori, A., Semisimple algebras over a commutative ring, J. Math. Soc. Japan, 15 (1963), 404–419.Google Scholar
[8] Hattori, A., Semisimple algebras over a commutative ring, Proc. Symp. Algebra. Math. Soc. Japan, 6 (1965), 37–40 (In Japanese).Google Scholar
[9] Hattori, A., Simple algebras over a commutative ring, To appear.Google Scholar
[10] Kanzaki, T., On commutor rings and Galois theory of separable algebras, Osaka Math. J., 1 (1964), 103–115.Google Scholar
[11] Watanabe, Y., Simple algebras over a complete local ring, Osaka Math. J., 3 (1966), 13–20.Google Scholar