A dimension theorem for division rings

LetD be a division algebra over a fieldk, letn be an arbitrary positive integer, and letk(x ₁,...,x _n) denote the rational function field inn variables overk. In this note we complete previous work by proving that the following three conditions are equivalent: (i) there exists an integerj such that the matrix ringM _j(D) contains a commutative subfield which has transcendence degreen overk; (ii) K dim (D⊗_k k(x ₁,...,x _n )) =n; (iii) gl. dim (D⊗_k k(x ₁,...,x _n )) =n. The crucial tool in the proof of this theorem is the Nullstellensatz forDx ₁,...,x _n] which was obtained by Amitsur and Small.