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Proceedings of the American Mathematical Society

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



Prime generalized alternative rings $ I$ with nontrivial idempotent

Author: Harry F. Smith
Journal: Proc. Amer. Math. Soc. 39 (1973), 242-246
MSC: Primary 17D05
MathSciNet review: 0313348
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Abstract: A generalized alternative ring $ I$ is a nonassociative ring $ R$ in which the identities $ (wx,y,z) + (w,x,[y,z]) - w(x,y,z) - (w,y,z)x;([w,x],y,z) + (w,x,yz) - y(w,x,z) - (w,x,y)z$; and $ (x,x,x)$ are identically zero. It is demonstrated here that if $ R$ is a ring of this type with characteristic different from two and three, then $ R$ semiprime with idempotent $ e$ implies that $ R$ has a Peirce decomposition relative to $ e$. Furthermore, if $ R$ is prime and $ e \ne 0,1$; then $ R$ must be alternative.

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Keywords: Generalized alternative ring $ I$, semiprime ring, Peirce decomposition, prime ring, alternative ring
Article copyright: © Copyright 1973 American Mathematical Society