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A commutativity theorem for division rings

Published online by Cambridge University Press:  17 April 2009

Hazar Abu-Khuzam  and
Adil Yaqub
Hazar Abu-Khuzam
Affiliation:
Department of Mathematics, University of California, Santa Barbara, California 93106, USA.
Adil Yaqub
Affiliation:
Department of Mathematics, University of California, Santa Barbara, California 93106, USA.
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Abstract

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The following theorem is proved: Let D be a division ring such that for all x, y in D there exists a positive integer n = n(x, y) for which (xy)n − (yx)n is in the center of D. Then D is commutative. This theorem also holds for semisimple rings.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 21 , Issue 1 , February 1980 , pp. 43 - 46
Copyright
Copyright © Australian Mathematical Society 1980

References

[1]Herstein, Israel N., “Sugli anelli soddisfacenti ad una condizione de Engel”, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 32 (1962), 177–180.Google Scholar
[2]Herstein, I.N., “A commutativity theorem”, J. Algebra 38 (1976), 112–118.CrossRefGoogle Scholar