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

 

On generalizing alternative rings


Author: D. J. Rodabaugh
Journal: Proc. Amer. Math. Soc. 46 (1974), 157-163
MSC: Primary 17D05
MathSciNet review: 0349786
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Abstract: Consider a ring $ R$ that satisfies the identity $ (x,x,x) = 0$ and any two of the three identities: $ (wx,y,z) + (w,x,[y,z]) - w(x,y,z) - (w,y,z)x = 0;([w,x],y,z) + (w,x,yz) - y(w,x,z) - (w,x,y)z = 0;(w,x \cdot y,z) - x \cdot (w,y,z) - y \cdot (w,x,z) = 0$. In this paper, we prove that if $ R$ has characteristic prime to 6 then $ R$ semiprime with idempotent $ e$ implies $ R$ has a Peirce decomposition in which the modules multiply as they do in an alternative ring. If in addition $ R$ is prime with idempotent $ e \ne 0,1$ then $ R$ is alternative.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1974-0349786-X
PII: S 0002-9939(1974)0349786-X
Keywords: Alternative ring, prime ring, semiprime ring
Article copyright: © Copyright 1974 American Mathematical Society