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Order relation in quadratic Jordan rings and a structure theorem


Authors: Santos González and Consuelo Martínez
Journal: Proc. Amer. Math. Soc. 104 (1988), 51-54
MSC: Primary 17C10
DOI: https://doi.org/10.1090/S0002-9939-1988-0958042-8
MathSciNet review: 958042
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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0958042-8
Keywords: Quadratic Jordan algebras, partial order, direct product, nilpotent element, hyperatomic, orthogonally complete
Article copyright: © Copyright 1988 American Mathematical Society

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