Extending a Jordan ring homomorphism

Abstract:

In this paper a homomorphism from an ideal $\mathcal {B}$ of a quadratic Jordan algebra $\mathcal {J}$ without $2$-torsion over a ring $\Phi$ onto a unital quadratic Jordan algebra $\mathcal {J}’$ without $2$-torsion is extended to a homomorphism from $\mathcal {J}$ to $\mathcal {J}’$. We then show if $D$ is any class of quadratic Jordan algebras without $2$-torsion, then the upper radical property determined by $D$ is hereditary.