A non-commutative Nullstellensatz

TL;DR

This paper establishes a Nullstellensatz variant for polynomial maps over finite-dimensional central division algebras, generalizing known results in commutative and quaternionic cases.

Contribution

It introduces a non-commutative Nullstellensatz for 2-sided ideals in polynomial rings over division algebras, extending classical and quaternionic Nullstellensatz results.

Findings

Proves a Nullstellensatz for polynomial maps over division algebras

Generalizes classical Nullstellensatz to non-commutative setting

Recovers known results for commutative fields and quaternions

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.