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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

[1]Abu-Khuzam, H., Tominaga, H., and Yaqub, A., ‘Commutativity theorems for s–unital rings satisfying polynomial identities’, Math. J. Okayama University 22 (1980), 111–114.Google Scholar
[2]Abu-Khuzam, H. and Yaqub, A., ‘n–torsion-free rings with commuting powers’, Math. Japon. 25 (1980), 37–42.Google Scholar
[3]Abu-Khuzam, H., ‘A commutativity theorem for periodic rings’, Math. Japon. 32 (1987), 1–3.Google Scholar
[4]Bell, H. E., ‘Some commutativity results for periodic rings’, Acta Math. Acad. Sci. Hungar. 28 (1976), 279–283.CrossRefGoogle Scholar
[5]Bell, H. E., ‘On rings with commuting powers’, Math. Japon. 24 (1979), 473–478.Google Scholar
[6]Bell, H. E., ‘On commutativity of periodic rings and near-rings’, Acta Math. Acad. Sci. Hungar. 36 (1980), 293–302.CrossRefGoogle Scholar
[7]Belluce, L. P., Herstein, I. N., and Jain, S. K., ‘Generalized commutative rings’, Nagoya Math. J. 27 (1966), 1–5.CrossRefGoogle Scholar
[8]Herstein, I. N., ‘A commutativity theorem’, J. Algebra 38 (1976), 112–118.CrossRefGoogle Scholar
[9]Ikehata, S. and Tominaga, H., ‘A commutativity theorem’, Math. Japon. 24 (1979), 29–30.Google Scholar
[10]Nicholson, W. K. and Yaqub, A., ‘A commutativity theorem for rings and groups’, Canad. Math. Bull. 22 (1979), 419–423.CrossRefGoogle Scholar
[11]Nicholson, W. K. and Yaqub, A., ‘A commutativity theorem’, Algebra Universalis 10 (1980), 260–263.CrossRefGoogle Scholar