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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Rings with associators in the commutative center


Author: Erwin Kleinfeld
Journal: Proc. Amer. Math. Soc. 104 (1988), 10-12
MSC: Primary 17A30
MathSciNet review: 958033
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Abstract: Thedy has introduced the subject of rings which satisfy the identity

$\displaystyle {\text{(I) }}(R,(R,R,R)) = 0,$

where the commutator is defined by $ (a,b) = ab - ba$, and the associator is defined by $ (a,b,c) = ab \cdot c - a \cdot bc$ and which satisfy one additional identity such as $ (x,x,x) = 0$. Assuming characteristic $ \ne 2$ and simplicity, Thedy's result is that such a ring $ R$ must be either commutative or associative. Thedy also showed that simplicity could not be relaxed to prime by presenting some examples which are neither associative nor commutative. We show here that the additional identity assumed by Thedy is in fact unnecessary for we show that if $ R$ is simple, characteristic $ \ne 2,3$ and satisfies (I) then $ R$ must be either commutative or associative.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0958033-7
PII: S 0002-9939(1988)0958033-7
Article copyright: © Copyright 1988 American Mathematical Society