A non-commutative Nullstellensatz

Abstract

Let $K$ be a field and $D$ be a finite-dimensional central division algebra over $K$. We prove a variant of the Nullstellensatz for $2$-sided ideals in the ring of polynomial maps $D^n\to D$. In the case where $D=K$ is commutative, our main result reduces to the $K$-Nullstellensatz of Laksov and Adkins-Gianni-Tognoli. In the case, where $K=\mathbb{R}$ is the field of real numbers and $D$ is the algebra of Hamilton quaternions, it reduces to the quaternionic Nullstellensatz recently proved by Alon and Paran.

7 pages