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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Quadratic Jordan algebras whose elements are all invertible or nilpotent
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by Kevin McCrimmon
Proc. Amer. Math. Soc. 35 (1972), 309-316
DOI: https://doi.org/10.1090/S0002-9939-1972-0308217-4

Abstract:

We prove that if $\mathfrak {J}$ is a unital quadratic Jordan algebra whose elements are all either invertible or nilpotent, then modulo the nil radical $\mathfrak {N}$ the algebra $\mathfrak {J}/\mathfrak {N}$ is either a division algebra or the Jordan algebra determined by a traceless quadratic form in characteristic 2. We also show that if $\mathfrak {U}$ is an associative algebra with involution whose symmetric elements are either invertible or nilpotent, then modulo its radical $\mathfrak {U}/\Re$ is a division algebra, a direct sum of anti-isomorphic division algebras, or a split quaternion algebra.
References
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Bibliographic Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 35 (1972), 309-316
  • MSC: Primary 17A15
  • DOI: https://doi.org/10.1090/S0002-9939-1972-0308217-4
  • MathSciNet review: 0308217