Quadratic Jordan algebras whose elements are all invertible or nilpotent

Author:
Kevin McCrimmon

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

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Abstract: We prove that if is a unital quadratic Jordan algebra whose elements are all either invertible or nilpotent, then modulo the nil radical the algebra is either a division algebra or the Jordan algebra determined by a traceless quadratic form in characteristic 2. We also show that if is an associative algebra with involution whose symmetric elements are either invertible or nilpotent, then modulo its radical is a division algebra, a direct sum of anti-isomorphic division algebras, or a split quaternion algebra.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1972-0308217-4

Keywords:
Quadratic Jordan algebra,
invertible element,
nilpotent element

Article copyright:
© Copyright 1972
American Mathematical Society