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On the Commutativity of CertainDivision Rings

Published online by Cambridge University Press:  20 November 2018

Tadasi Nakayama
Tadasi Nakayama*
Affiliation:
Nagoya University
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Formerly Hua [1] proved that if A is a division ring with centre Z and if there exists a natural number n such that an ∈ Z for every a ∈ A, then A is commutative; this generalizes Wedderburn's theorem on finite division rings. Another generalization of Wedderburn's theorem, due to Jacobson [3], asserts that every algebraic division algebra over a finite field is commutative. On the other hand, a theorem of Noether and Jacboson [3] states that every noncommutative algebraic division algebra contains an element which is not contained in the centre Z and is separable over Z.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 5 , 1953 , pp. 242 - 244
Copyright
Copyright © Canadian Mathematical Society 1953

References

1. Hua, L. K., Some properties of a sfield, Proc. Nat. Acad. Sci. U.S.A., 35 (1949), 533–537.Google Scholar
2. Ikeda, M., On a theorem of Kaplansky, Osaka Math. J., 4 (1952), 235–240.Google Scholar
3. Jacobson, N., Structure theory of algebraic algebras of bounded degree, Ann. Math., 46 (1945), 695–707.Google Scholar
4. Kaplansky, I., A theorem on division rings, Can. J. Math., 3 (1951), 290–292.Google Scholar
5. Krasner, M., On the non-existence of certain extensions, Amer. J. Math., 75 (1953), 112–116.Google Scholar