Local orders in Jordan algebras. (English) Zbl 1417.16025
Summary: Making use of results on general algebras of quotients of Jordan algebras, we study a notion of local order based on the version for linear Jordan algebras of the ideas of J. Fountain and V. Gould [Commun. Algebra 18, No. 9, 3085–3110 (1990; Zbl 0719.16022)] as adapted to the Jordan context by A. Fernández-López and E. García-Rus in [J. Algebra 174, 1024–1048 (1995; Zbl 0830.17015)]. 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 Fernández-López and García-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.
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