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

Authors: Antonio Fernández López and Eulalia García Rus
Journal: Proc. Amer. Math. Soc. 108 (1990), 69-71
MSC: Primary 17C10; Secondary 16A34, 17C65
MathSciNet review: 991702
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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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