A characterization of the elements of the socle of a Jordan algebra

Fernández López, Antonio; García Rus, Eulalia

doi:10.1090/S0002-9939-1990-0991702-0

A characterization of the elements of the socle of a Jordan algebra
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by Antonio Fernández López and Eulalia García Rus

Proc. Amer. Math. Soc. 108 (1990), 69-71

DOI: https://doi.org/10.1090/S0002-9939-1990-0991702-0

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Abstract:

Let $J$ be a nondegenerate Jordan algebra over a field $K$ of characteristic not 2 . Here we prove that an element $b \in J$ is in the socle if and only if $J$ satisfies dcc on all principal inner ideals ${U_y}J,y \in Kb + {U_b}J$. By using this result we show that the socle of a quadratic extension ${J_F}$ of $J$ coincides with the quadratic extension ${\text {Soc(}}J{)_F}$ of its socle.

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