Quadratic Jordan algebras whose elements are all regular or nilpotent

Author:
Kevin McCrimmon

Journal:
Proc. Amer. Math. Soc. **45** (1974), 19-27

MSC:
Primary 17A15

DOI:
https://doi.org/10.1090/S0002-9939-1974-0374202-1

MathSciNet review:
0374202

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Abstract: We prove that if is a quadratic Jordan algebra whose elements are all either regular or nilpotent, and which satisfies a common multiple property (that whenever is nilpotent and regular then ), then modulo the radical the algebra is either a domain or a form of a Jordan algebra determined by a traceless quadratic form in characteristic 2.

**[1]**N. Jacobson,*Lectures on quadratic Jordan algebras*, Tata Institute Lecture Notes, Bombay, 1970. MR**0325715 (48:4062)****[2]**K. McCrimmon,*Quadratic Jordan algebras whose elements are all invertible or nilpotent*, Proc. Amer. Math. Soc.**35**(1972), 309-316. MR**0308217 (46:7332)****[3]**M. S. Montgomery,*Rings of quotients for a class of special Jordan rings*, J. Algebra (to appear). MR**0374203 (51:10403)****[4]**-,*Rings with involution in which every trace is nilpotent or regular*, Canad. J. Math. (to appear). MR**0330214 (48:8552)****[5]**M. Chacron and J. Chacron,*Rings with involution all of whose symmetric elements are nilpotent or regular*, Proc. Amer. Math. Soc.**37**(1973), 397-402. MR**0320058 (47:8599)****[6]**P. M. Cohn,*Prime rings with involution whose symmetric zero divisors are nilpotent*, Proc. Amer. Math. Soc. (to appear). MR**0318202 (47:6749)**

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DOI:
https://doi.org/10.1090/S0002-9939-1974-0374202-1

Article copyright:
© Copyright 1974
American Mathematical Society