Amitsur-Small rings

Let R n = D [ x 1 , … , 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 .