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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A characterisation of von Neumann regular Jordan triple systems


Author: Kurt Meyberg
Journal: Proc. Amer. Math. Soc. 49 (1975), 25-27
MSC: Primary 17C10
DOI: https://doi.org/10.1090/S0002-9939-1975-0376795-8
MathSciNet review: 0376795
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Abstract: In this note we give a characterisation of the Jacobson radical of a Jordan triple system in terms of principal inner ideals. If $ (\mathfrak{A},P)$ is a Jordan triple system and $ \operatorname{Rad} \mathfrak{A}$ the Jacobson radical of $ \mathfrak{A}$ then $ x \in \operatorname{Rad} \mathfrak{A}$ iff $ P(x)\mathfrak{A} = P(x + P(x)y)\mathfrak{A}$ for all $ y \in \mathfrak{A}$. We use this to give a new characterisation of von Neumann regular Jordan triple systems. In particular, this gives another most elementary and short proof that semi-simple Jordan triple systems with dcc on principal inner ideals are von Neumann regular.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0376795-8
Keywords: Jordan algebra, Jordan triple system, Jacobson radical, von Neumann regular
Article copyright: © Copyright 1975 American Mathematical Society