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Skew fields with a non-trivial generalised power central rational identity

Published online by Cambridge University Press:  17 April 2009

Katsuo Chiba
Katsuo Chiba
Affiliation:
Niihama National College of Technology, 7-1 Yagumo-Cho, Niihama 792, Japan
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Abstract

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Let D be a skew field with uncountable centre K. The main result in the present paper is as follows: If D satisfies a non-trivial generalised power central rational identity, then D is finite dimensional over K. As a corollary we obtain the following result. Let a be an element of D such that (a−1x−1ax)q(x) ∈ K for all x ∈ D ﹨ {0} where q(x) is a positive integer depending on x. Then a ∈ K.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 49 , Issue 1 , February 1994 , pp. 85 - 90
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Amisur, S.A., ‘Generalized polynomial identities and pivotal monomials’, Trans. Amer. Math. Soc. 114 (1965), 210–226.Google Scholar
[2]Amisur, S.A., ‘Rational identities and applications to algebra and geometry’, J. Algebra 3 (1966), 304–359.Google Scholar
[3]Bergman, G.M., ‘The diamond lemma for ring theory’, Adv. in Math. 29 (1978), 178–218.Google Scholar
[4]Cohn, P.M., Free rings and their relations (Academic Press, New York and London, 1971).Google Scholar
[5]Cohn, P.M., Skew field constructions, London Math. Soc. Lecture Notes No. 27 (Cambridge University Press, Cambridge, 1977).Google Scholar
[6]Goncalves, J.Z., ‘Free groups in subnormal subgroups and the residual nilpotence of the groups of units of groups rings’, Canad. Math. Bull. 27 (1984), 365–370.Google Scholar
[7]Herstein, I.N., ‘Multiplicative commutators in division rings’, Israel J. Math. 31 (1978), 180–188.CrossRefGoogle Scholar
[8]Hetstein, I.N., ‘Multicative commutators in division rings II’, Rend. Circ. Mat. Palermo II 29 (1980), 485–489.Google Scholar
[9]Lewin, J., ‘Fields of fractions for group algebras of free groups’, Trans. Amer. Math. Soc. 192 (1974), 339–346.Google Scholar
[10]Makar-Limanov, L. and Malcolmson, P., ‘Words periodic over the center of a division ring’, Proc. Amer. Math. Soc. 93 (1985), 590–592.Google Scholar