Let I be a proper left ideal in the ring H[x1,…,xn] of polynomials in n central variables over the quaternion algebra H. Then there exists a point a=(a1,…,an)∈Hn with aiaj=ajai 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,…,xn] 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.
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- Polynomial rings