On the radical of a free Malcev algebra

Abstract:

We prove that the prime radical $rad \mathcal {M}$ of the free Malcev algebra $\mathcal {M}$ of rank more than two over a field of characteristic $\neq 2$ coincides with the set of all universally Engelian elements of $\mathcal {M}$. Moreover, let $T(\mathbb M)$ be the ideal of $\mathcal {M}$ consisting of all stable identities of the split simple 7-dimensional Malcev algebra $\mathbb M$ over $F$. It is proved that $rad \mathcal {M}=J(\mathcal {M})\cap T(\mathbb M)$, where $J(\mathcal {M})$ is the Jacobian ideal of $\mathcal {M}$. Similar results were proved by I. Shestakov and E. Zelmanov for free alternative and free Jordan algebras.