A non-commutative Nullstellensatz

Abstract

Let [Formula: see text] be a field and [Formula: see text] be a finite-dimensional central division algebra over [Formula: see text]. We prove a variant of the Nullstellensatz for [Formula: see text]-sided ideals in the ring of polynomial maps [Formula: see text]. In the case where [Formula: see text] is commutative, our main result reduces to the [Formula: see text]-Nullstellensatz of Laksov and Adkins–Gianni–Tognoli. In the case, where [Formula: see text] is the field of real numbers and [Formula: see text] is the algebra of Hamilton quaternions, it reduces to the quaternionic Nullstellensatz recently proved by Alon and Paran.