A central quaternionic Nullstellensatz

Let I be a proper left ideal in the ring H[x₁,…,x_n] of polynomials in n central variables over the quaternion algebra H. Then there exists a point a=(a₁,…,a_n)∈Hⁿ with a_ia_j=a_ja_i for all i,j, such that every polynomial in I vanishes at a. This generalizes a theorem of Jacobson, who proved the case n=1. Moreover, a polynomial f∈H[x₁,…,x_n] vanishes at all common zeroes of polynomials in I if and only if f belongs to the intersection of all completely prime left ideals that contain I – a notion introduced by Reyes in 2010.