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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A generalization of commutative and alternative rings. IV


Author: Erwin Kleinfeld
Journal: Proc. Amer. Math. Soc. 46 (1974), 21-23
MSC: Primary 17D05
DOI: https://doi.org/10.1090/S0002-9939-1974-0424889-X
MathSciNet review: 0424889
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Abstract: We have shown previously for rings $ R$ of characteristic $ \ne 2,3$ which satisfy the three identities (i) $ (x,{y^2},x) = y \circ (x,y,x)$, (ii) $ (x,y,z) + (y,z,x) + (z,x,y) = 0$, and (iii) ( $ ((x,y),x,x) = 0$, where $ (a,b,c) = (ab)c - a(bc),(a,b) = ab - ba$, and $ a \circ b = ab + ba$, that under the assumption of no divisors of zero, all such $ R$ must be either associative or commutative. Here we weaken the Lie-admissible identity (ii) by assuming instead (iv) Lie-admissibility for every subring generated by two elements. It turns out that rings without divisors of zero and of characteristic $ \ne 2,3$ which satisfy (i), (iii) and (iv) are either commutative or alternative. If $ S$ is a ring in which every subring generated by two elements is either commutative or associative, then identities (i), (iii) and (iv) hold in $ S$, so that this result applies to $ S$.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0424889-X
Keywords: Commutative, alternative, flexible, noncommutative, Jordan, zero divisor
Article copyright: © Copyright 1974 American Mathematical Society

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