Primitive noncommutative Jordan algebras with nonzero socle

Authors:
Antonio Fernandez Lopez and Angel Rodriguez Palacios

Journal:
Proc. Amer. Math. Soc. **96** (1986), 199-206

MSC:
Primary 17A15; Secondary 16A68

DOI:
https://doi.org/10.1090/S0002-9939-1986-0818443-3

MathSciNet review:
818443

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $A$ be a nondegenerate noncommutative Jordan algebra over a field $K$ of characteristic $\ne 2$. Defining the socle $S(A)$ of $A$ to be the socle of the plus algebra ${A^ + }$, we prove that $S(A)$ is an ideal of $A$; then we prove that if $A$ has nonzero socle, $A$ is prime if and only if it is primitive, extending a result of Osborn and Racine [**6**] for the associative case. We also describe the prime noncommutative Jordan algebras with nonzero socle and in particular the simple noncommutative Jordan algebras containing a completely primitive idempotent. In fact we prove that a nondegenerate prime noncommutative Jordan algebra with nonzero socle is either (i) a noncommutative Jordan division algebra, (ii) a simple flexible quadratic algebra over an extension of the base field, (iii) a nondegenerate prime (commutative) Jordan algebra with nonzero socle, or (iv) a $K$-subalgebra of ${L_W}{(V)^{(\lambda )}}$ containing ${F_W}(V)$ or of $H{({L_V}(V), * )^{(\lambda )}}$ containing $H({F_V}(V), * )$ where in the first case $(V,W)$ is a pair of dual vector spaces over an associative division $K$-algebra $D$ and $\lambda \ne 1/2$ is a central element of $D$, and where in the second case $V$ is self-dual with respect to an hermitian inner product $(|),D$ has an involution $\alpha \to \bar \alpha$ and $\lambda \ne 1/2$ is a central element of $D$ with $\lambda + \bar \lambda = 1$.

- A. A. Albert,
*Power-associative rings*, Trans. Amer. Math. Soc.**64**(1948), 552–593. MR**27750**, DOI https://doi.org/10.1090/S0002-9947-1948-0027750-7 - Leslie Hogben and Kevin McCrimmon,
*Maximal modular inner ideals and the Jacobson radical of a Jordan algebra*, J. Algebra**68**(1981), no. 1, 155–169. MR**604300**, DOI https://doi.org/10.1016/0021-8693%2881%2990291-X - Nathan Jacobson,
*Structure of rings*, Revised edition, American Mathematical Society Colloquium Publications, Vol. 37, American Mathematical Society, Providence, R.I., 1964. MR**0222106** - Nathan Jacobson,
*Structure and representations of Jordan algebras*, American Mathematical Society Colloquium Publications, Vol. XXXIX, American Mathematical Society, Providence, R.I., 1968. MR**0251099** - Kevin McCrimmon,
*Noncommutative Jordan rings*, Trans. Amer. Math. Soc.**158**(1971), 1–33. MR**310024**, DOI https://doi.org/10.1090/S0002-9947-1971-0310024-7 - J. Marshall Osborn and M. L. Racine,
*Jordan rings with nonzero socle*, Trans. Amer. Math. Soc.**251**(1979), 375–387. MR**531985**, DOI https://doi.org/10.1090/S0002-9947-1979-0531985-4 - Richard D. Schafer,
*An introduction to nonassociative algebras*, Pure and Applied Mathematics, Vol. 22, Academic Press, New York-London, 1966. MR**0210757** - M. Slater,
*The socle of an alternative ring*, J. Algebra**14**(1970), 443–463. MR**260817**, DOI https://doi.org/10.1016/0021-8693%2870%2990094-3 - Kirby C. Smith,
*Noncommutative Jordan algebras of capacity two*, Trans. Amer. Math. Soc.**158**(1971), 151–159. MR**277584**, DOI https://doi.org/10.1090/S0002-9947-1971-0277584-6 - E. I. Zel′manov,
*Jordan division algebras*, Algebra i Logika**18**(1979), no. 3, 286–310, 385 (Russian). MR**566787**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
17A15,
16A68

Retrieve articles in all journals with MSC: 17A15, 16A68

Additional Information

Article copyright:
© Copyright 1986
American Mathematical Society