Polynomial identities and speciality of Martindale-like covers of Jordan algebras

Abstract

In this paper we prove the inheritance of polynomial identities by covers of nondegenerate Jordan algebras satisfying certain ideal absorption properties. As a consequence we obtain the inheritance of speciality by Martindale-like covers, proving, in particular, that a Jordan algebra having a nondegenerate essential ideal which is special must be special.

Algebras of quotients of Jordan algebras

“…We refer to [GG,AGG] for the notion of Martindale-like algebra of quotients of a linear Jordan algebra, which has been generalized for quadratic Jordan algebras to the notion of Martindale-like cover [ACGG1,ACGG2]. Let J be a Jordan algebra and let F be a filter of essential ideals of J satisfying the property: for all I ∈ F , the derived ideal I (1) = U I I is again in F .…”

“…It was proved in [ACGG1] that if Q is a Martindale-like cover of a nondegenerate Jordan algebra J , then: (a) if J is PI, then Q is PI, and in this case, every homogeneous polynomial p which vanishes on J , also vanishes on Q [ACGG1,2.5], and (b) if J is special, then Q is special. It is clear that the proof of 3.8(1) adapts to yield the corresponding results for any algebra of quotients Q of a nondegenerate J , which contains the above results as particular cases.…”

“…It is clear that the proof of 3.8(1) adapts to yield the corresponding results for any algebra of quotients Q of a nondegenerate J , which contains the above results as particular cases. Of course, the situation in [ACGG1] is in principle more general, since the authors consider in that work what they call covers satisfying the condition C w ( J ) ⊆ C w (Q ), and the outer absorption property IA1 of [ACGG1, 0.10]: for any q ∈ Q there exists an essential ideal I of J such that 0 = U I q ∈ J . Note however the following Lemma.…”

“…Let Q be a cover of the nondegenerate Jordan algebra J . If J is PI and Q satisfies IA1 of [ACGG1], then Q is a Martindale-like cover of J .…”

Martindale quotients of Jordan algebras

“…We impose no conditions (such as semiprimeness or nondegeneracy), only that the ''denominators'' are faithful to J (sturdy). This notion extends that given in the linear setting by García and Gómez-Lozano [3], and also includes the notion of Martindale-like cover [1,2] for nondegenerate algebras. Since we do not assume any regularity condition other than the existence of a denominator filter of ideals, we cannot make use of the structure theory of nondegenerate Jordan algebras, unlike [1][2][3]14].…”