## 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
M. Benslimane, - 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**

*Charactérisations des algébres de Jordan Banach non commutatives de capacité finie et caractérisation spectrale des algébres de Jordan Banach non commutatives modulaires annihilatrices*, Thesis, Université de Rabat, 1987.

## Additional Information

- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**108**(1990), 69-71 - MSC: Primary 17C10; Secondary 16A34, 17C65
- DOI: https://doi.org/10.1090/S0002-9939-1990-0991702-0
- MathSciNet review: 991702