Order relation in quadratic Jordan rings and a structure theorem

González, Santos; Martínez, Consuelo

doi:10.1090/S0002-9939-1988-0958042-8

Order relation in quadratic Jordan rings and a structure theorem
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by Santos González and Consuelo Martínez

Proc. Amer. Math. Soc. 104 (1988), 51-54

DOI: https://doi.org/10.1090/S0002-9939-1988-0958042-8

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Abstract:

It is shown that the relation defined by $x \leq y$ if and only if ${V_x}x = {V_x}y$ and ${U_x}x = {U_x}y = {U_y}x$ is an order relation for quadratic Jordan algebras without nilpotent elements, which extends our previous one for linear Jordan algebras, and reduces to the usual Abian order for associative algebras. We prove that a quadratic Jordan algebra is isomorphic to a direct product of division algebras if and only if the algebra has no nilpotent elements and is hyperatomic and orthogonally complete.

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