Amitsur-Small rings

Let R_n=D[x₁,…,x_n] denote the ring of polynomials in n central variables over a division ring D. We say that D is an Amitsur-Small ring if for any maximal left ideal in R_n, M∩R_k is a maximal left ideal in R_k, for all n∈N and 1≤k≤n. We demonstrate the existence of non Amitsur-Small division rings, providing a negative answer to a question of Amitsur and Small from 1978. We show that Hamilton's real quaternion algebra H=(−1,−1)_2,R is an Amitsur-Small ring, division rings of degree 3 over their center F are never Amitsur-Small, and division rings of degree 2 are not Amitsur-Small if they are not quaternion algebras (−1,−1)_2,F over a Pythagorean field F.