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

Semiprimeness of special Jordan algebras


Author: Kevin McCrimmon
Journal: Proc. Amer. Math. Soc. 96 (1986), 29-33
MSC: Primary 17C10; Secondary 16A68
MathSciNet review: 813803
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Abstract: There are important connections between radicals of a special Jordan algebra $ J$ and its associative envelope $ A$. For the locally nilpotent (Levitzki) radical $ \mathcal{L}$, Skosyrskii proved $ \mathcal{L}(J) = J \cap \mathcal{L}(A)$. For the prime (Baer) radical $ \mathcal{P}$, Erickson and Montgomery proved $ \mathcal{P}(J) = J \cap \mathcal{P}(A)$ when $ J = H(A, * )$ consists of all symmetric elements of an algebra $ A$ with involution $ * $. In his important work on prime Jordan algebras, Zelmanov proved $ \mathcal{P}(J) = J \cap \mathcal{P}(A)$ for all linear $ J$ and all associative envelopes $ A$. In the present paper we extend Zelmanov's result to arbitrary quadratic Jordan algebras. In particular, we see that a special Jordan algebra is semiprime iff it has some semiprime associative envelope.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0813803-9
PII: S 0002-9939(1986)0813803-9
Keywords: Special Jordan algebra, semiprime, associative enveloping algebra
Article copyright: © Copyright 1986 American Mathematical Society