Amitsur-Small rings

Let $R_n=D[x_1,\dots ,x_n]$ 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 $R_n$, $M\cap R_k$ is a maximal left ideal in $R_k$, for all $n\in N$ and $1\le k\le n$. 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.