Primitive Noncommutative Jordan Algebras with Nonzero Socle

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 $(|),D$ has an involution $\alpha \to \bar \alpha$ and $\lambda \ne 1/2$ is a central element of $D$ with $\lambda + \bar \lambda = 1$.