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Proceedings of the American Mathematical Society

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

 

 

Primitive noncommutative Jordan algebras with nonzero socle


Authors: Antonio Fernandez Lopez and Angel Rodriguez Palacios
Journal: Proc. Amer. Math. Soc. 96 (1986), 199-206
MSC: Primary 17A15; Secondary 16A68
MathSciNet review: 818443
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Abstract: Let $ A$ be a nondegenerate noncommutative Jordan algebra over a field $ K$ of characteristic $ \ne 2$. Defining the socle $ S(A)$ of $ A$ to be the socle of the plus algebra $ {A^ + }$, we prove that $ S(A)$ is an ideal of $ A$; then we prove that if $ A$ has nonzero socle, $ A$ is prime if and only if it is primitive, extending a result of Osborn and Racine [6] for the associative case. We also describe the prime noncommutative Jordan algebras with nonzero socle and in particular the simple noncommutative Jordan algebras containing a completely primitive idempotent. In fact we prove that a nondegenerate prime noncommutative Jordan algebra with nonzero socle is either (i) a noncommutative Jordan division algebra, (ii) a simple flexible quadratic algebra over an extension of the base field, (iii) a nondegenerate prime (commutative) Jordan algebra with nonzero socle, or (iv) a $ K$-subalgebra of $ {L_W}{(V)^{(\lambda )}}$ containing $ {F_W}(V)$ or of $ H{({L_V}(V), * )^{(\lambda )}}$ containing $ H({F_V}(V), * )$ where in the first case $ (V,W)$ is a pair of dual vector spaces over an associative division $ K$-algebra $ D$ and $ \lambda \ne 1/2$ is a central element of $ D$, and where in the second case $ V$ is self-dual with respect to an hermitian inner product $ (\vert),D$ has an involution $ \alpha \to \bar \alpha $ and $ \lambda \ne 1/2$ is a central element of $ D$ with $ \lambda + \bar \lambda = 1$.


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DOI: https://doi.org/10.1090/S0002-9939-1986-0818443-3
Article copyright: © Copyright 1986 American Mathematical Society