A central quaternionic Nullstellensatz

• DOI: 10.1016/j.jalgebra.2021.01.018
• Publication Date: 2021-01-30T11:33:47Z
• Authors: Gil Alon; Elad Paran
• Keywords: 0101 mathematics; 01 natural sciences

Abstract Let I be a proper left ideal in the ring H [ x 1 , … , x n ] of polynomials in n central variables over the quaternion algebra H . Then there exists a point a = ( a 1 , … , a n ) ∈ H n with a i a j = a j a 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 [ x 1 , … , 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.