Local orders in Jordan algebras

TL;DR

This paper explores the concept of order in Jordan algebras, characterizing Lesieur-Croisot elements via algebra of quotients, and extends local order results to quadratic Jordan algebras with specific chain conditions.

Contribution

It introduces a Jordan algebra-specific notion of order, characterizes Lesieur-Croisot elements, and extends local order theory to quadratic Jordan algebras under certain conditions.

Findings

Lesieur-Croisot elements lie in the socle of the maximal algebra of quotients.

Characterization of local orders in nondegenerate Jordan algebras.

Extension of local order results to quadratic Jordan algebras with chain conditions.

Abstract

We study a notion of order in Jordan algebras based on the version for Jordan algebras of the ideas of Fountain and Gould as adapted to the Jordan context by Fern\'{a}ndez-L\'{o}pez and Garc\'{\i}a-Rus, making use of results on general algebras of quotients of Jordan algebras. In particular, we characterize the set of Lesieur-Croisot elements of a nondegenerate Jordan algebra as those elements of the Jordan algebra lying in the socle of its maximal algebra of quotients, and apply this relationship to extend to quadratic Jordan algebras the results of Fern\'{a}ndez-L\'{o}pez and Garc\'{\i}a-Rus on local orders in nondegenerate Jordan algebras satisfying the descending chain condition on principal inner ideals and not containing ideals which are nonartinian quadratic factors.