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Commutativity results for rings

Published online by Cambridge University Press:  17 April 2009

Hazar Abu-Khuzam
Hazar Abu-Khuzam
Affiliation:
Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
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Abstract

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Let R be an associative ring. We prove that if for each finite subset F of R there exists a positive integer n = n(F) such that (xy)n − yn xn is in the centre of R for every x, y in F, then the commutator ideal of R is nil. We also prove that if n is a fixed positive integer and R is an n(n + 1)-torsion-free ring with identity such that (xy)n − ynxn = (yx)n xnyn is in the centre of R for all x, y in R, then R is commutative.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 38 , Issue 2 , October 1988 , pp. 191 - 195
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Abu-Khuzam, H., ‘A commutativity theorem for rings’, Math. Japon. 25 (1980), 593–595.Google Scholar
[2]Bell, H.E., ‘On some commutativity theorems of Herstein’, Arch. Math. (Basel) 24 (1973), 34–38.CrossRefGoogle Scholar
[3]Bell, H.E., ‘On rings with commuting powers’, Math. Japon. 24 (1979), 473–478.Google Scholar
[4]Gupta, V., ‘Some remarks on the commutativity of rings’, Acta Math. Acad. Sci. Hungar 38 (1980), 233–236.CrossRefGoogle Scholar
[5]Herstein, I.N., ‘Power maps in rings’, Michigan Math. J. 8 (1961), 29–32.CrossRefGoogle Scholar
[6]Herstein, I.N., ‘A commutativity theorem’, J. Algebra 38 (1976), 112–118.CrossRefGoogle Scholar
[7]Hirano, Y., ‘Some polynomial identities and commutativity of rings II’, Proc. of the 14-th Symposium on Ring Theory, Okayama (1982), 9–24.Google Scholar
[8]Jacobson, N., Structure of Rings (A.M.S. Colloquium Publication, 1964).Google Scholar
[9]Kezlan, T.P., ‘A note on commutativity of semiprime PI-rings’, Math. Japon. 27 (1982), 267–268.Google Scholar
[10]Nicholson, W.K. and Yaqub, A., ‘A commutativity theorem’, Algebra Universalis 10 (1980), 260–263.CrossRefGoogle Scholar