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Commutativity of rings satisfying certain polynomial identities

Published online by Cambridge University Press:  17 April 2009

Hazar Abu-Khuzam ,
Howard Bell  and
Adil Yaqub
Hazar Abu-Khuzam
Affiliation:
Department of Mathematics, Kuwait University, Kuwait13060
Howard Bell
Affiliation:
Department of Mathematics, Brock University, St. Catharines, Ontario, CanadaL2S 3A1
Adil Yaqub
Affiliation:
Department of Mathematics, University of California, Santa Barbara, CA 93106, United States of America
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Abstract

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It is shown that an n-torsion-free ring R with identity such that, for all x, y in R, xnyn = ynyn and (xy)n+1 − xn+1yn+1 is central, must be commutative. It is also shown that a periodic n–torsion-free ring (not necessarily with identity) for which (xy)n − (yx)n is always in the centre is commutative provided that the nilpotents of R form a commutative set. Further, examples are given which show that all the hypotheses of both theorems are essential.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 44 , Issue 1 , August 1991 , pp. 63 - 69
Copyright
Copyright © Australian Mathematical Society 1991

References

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