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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On a generalization of alternative and Lie rings


Author: Erwin Kleinfeld
Journal: Trans. Amer. Math. Soc. 155 (1971), 385-395
MSC: Primary 17.10
DOI: https://doi.org/10.1090/S0002-9947-1971-0272839-3
MathSciNet review: 0272839
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Abstract: Alternative as well as Lie rings satisfy all of the following four identities: (i) $ ({x^2},y,z) = x(x,y,z) + (x,y,z)x$, (ii) $ (x,{y^2},z) = y(x,y,z) + (x,y,z)y$, (iii) $ (x,y,{z^2}) = z(x,y,z) + (x,y,z)z$, (iv) $ (x,x,x) = 0$, where the associator $ (a,b,c)$ is defined by $ (a,b,c) = (ab)c - a(bc)$. If $ R$ is a ring of characteristic different from two and satisfies (iv) and any two of the first three identities, then it is shown that a necessary and sufficient condition for $ R$ to be alternative is that whenever $ a,b,c$ are contained in a subring $ S$ of $ R$ which can be generated by two elements and whenever $ {(a,b,c)^2} = 0$, then $ (a,b,c) = 0$.


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DOI: https://doi.org/10.1090/S0002-9947-1971-0272839-3
Keywords: Alternative ring, nonassociative ring, identities, division ring, Artin's theorem, projective plane
Article copyright: © Copyright 1971 American Mathematical Society