Parallels between quaternionic and matrix Nullstellensätze

We prove a new quaternionic and a new matrix nullstellensatz. We also show that both theories are intertwined. For every g 1 , … , g m , f ∈ H [ x 1 , … , x d ] (where x 1 , … , x d are central) we show that the following are equivalent: (a) For every a ∈ H d whose components pairwise commute and which satisfies g 1 ( a ) = … = g m ( a ) = 0 we have f ( a ) = 0 . (b) f belongs to the smallest semiprime left ideal containing g 1 , … , g m . On the other hand, for every G 1 , … , G m , F ∈ M n ( k [ x 1 , … , x d ] ) , where k is an algebraically closed field, we show that the following are equivalent (where I is the left ideal generated by G 1 , … , G m ): (a) For every a ∈ k d and v ∈ k n such that G 1 ( a ) v = … = G m ( a ) v = 0 , we have F ( a ) v = 0 . (b) For every A ∈ M n ( k ) there exists N ∈ N 0 such that ( A F ) N ∈ I + I ( A F ) + … + I ( A F ) N .