Jordan centers and Martindale-like covers

Abstract

In this paper we show that the scalar center of a nondegenerate quadratic Jordan algebra is contained in the scalar center of any of its Martindale-like covers.

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 .…”

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].…”

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

“…When J and Q are Jordan algebras such that J is a subalgebra of Q, we will say that Q is a cover of J. Following [1], we will consider the following ideal absorption properties of a cover Q of J:…”

“…, a n ∈ J . We will say that p ∈ FJ (1) x (2) x (3) x (4) summed over all permutations on 4 letters, vanishes on any Albert form (cf. [10, p. 112]), hence, it vanishes strictly on any strongly prime exceptional Jordan algebra (see [11, 15.2]).…”

“…In this paper we present a different approach to that subject through polynomial identities, which allows us to solve the problem in the even more general setting of (quadratic) Martindale-like covers [1,Section 2]. On the other hand, we prove results of independent interest on the inheritance of polynomial identities.…”