A characterization of the elements of the socle of a Jordan algebra

Fernández López, Antonio; García Rus, Eulalia

doi:10.1090/S0002-9939-1990-0991702-0

A characterization of the elements of the socle of a Jordan algebra
HTML articles powered by AMS MathViewer

by Antonio Fernández López and Eulalia García Rus PDF

Proc. Amer. Math. Soc. 108 (1990), 69-71 Request permission

Abstract:

Let $J$ be a nondegenerate Jordan algebra over a field $K$ of characteristic not 2 . Here we prove that an element $b \in J$ is in the socle if and only if $J$ satisfies dcc on all principal inner ideals ${U_y}J,y \in Kb + {U_b}J$. By using this result we show that the socle of a quadratic extension ${J_F}$ of $J$ coincides with the quadratic extension ${\text {Soc(}}J{)_F}$ of its socle.

References

P. N. Ánh, Simple Jordan algebras with minimal inner ideals, Comm. Algebra 14 (1986), no. 3, 489–492. MR 823349, DOI 10.1080/00927878608823320

Charactérisations des algébres de Jordan Banach non commutatives de capacité finie et caractérisation spectrale des algébres de Jordan Banach non commutatives modulaires annihilatrices

Antonio Fernández López, Modular annihilator Jordan algebras, Comm. Algebra 13 (1985), no. 12, 2597–2613. MR 811525, DOI 10.1080/00927878508823292
Antonio Fernandez Lopez and Angel Rodriguez Palacios, Primitive noncommutative Jordan algebras with nonzero socle, Proc. Amer. Math. Soc. 96 (1986), no. 2, 199–206. MR 818443, DOI 10.1090/S0002-9939-1986-0818443-3
Antonio Fernández López and Ángel Rodríguez-Palacios, On the socle of a noncommutative Jordan algebra, Manuscripta Math. 56 (1986), no. 3, 269–278. MR 856365, DOI 10.1007/BF01180768
Nathan Jacobson, Structure and representations of Jordan algebras, American Mathematical Society Colloquium Publications, Vol. XXXIX, American Mathematical Society, Providence, R.I., 1968. MR 0251099
Ottmar Loos, Jordan pairs, Lecture Notes in Mathematics, Vol. 460, Springer-Verlag, Berlin-New York, 1975. MR 0444721
Kevin McCrimmon, Inner ideals in quadratic Jordan algebras, Trans. Amer. Math. Soc. 159 (1971), 445–468. MR 279145, DOI 10.1090/S0002-9947-1971-0279145-1
Kurt Meyberg, Lectures on algebras and triple systems, University of Virginia, Charlottesville, Va., 1972. Notes on a course of lectures given during the academic year 1971–1972. MR 0340353
J. Marshall Osborn and M. L. Racine, Jordan rings with nonzero socle, Trans. Amer. Math. Soc. 251 (1979), 375–387. MR 531985, DOI 10.1090/S0002-9947-1979-0531985-4
K. A. Zhevlakov, A. M. Slin′ko, I. P. Shestakov, and A. I. Shirshov, Rings that are nearly associative, Pure and Applied Mathematics, vol. 104, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. Translated from the Russian by Harry F. Smith. MR 668355