Multiplicative semiprimeness of strongly semiprime non-commutative Jordan algebras

Abstract

In this paper, we prove that the multiplication algebra of a nondegenerate non-commutative Jordan algebra is semiprime as a consequence of the multiplicative primeness of strongly prime non-commutative Jordan algebras, obtained previously by the two first named authors. For that we prove first the coincidence of the McCrimmon radicals for a non-commutative Jordan algebra and its symmetrization, which allows us to describe the nondegenerate ones as a subdirect product of a family of strongly prime algebras. Indeed, we obtain that the minimal nondegenerate ideals of the symmetrization of any flexible algebra are ideals of the original algebra. This is based on its invariance under the derivations of the algebra, which requires working on vector spaces over a field of 0 characteristic.