Division rings and $V$-domains

Abstract:

Let $D$ be a division ring with center $k$ and let $k\left ( x \right )$ denote the field of rational functions over $k$. A square matrix $\tau \in {M_n}\left ( D \right )$ is said to be totally transcendental over $k$ if the evaluation map $\varepsilon : k\left [ x \right ] \to {M_n}\left ( D \right ),\varepsilon \left ( f \right ) = f\left ( \tau \right )$, can be extended to $k\left ( x \right )$. In this note it is shown that the tensor product $D{ \otimes _k}k\left ( x \right )$ is a $V$-domain which has, up to isomorphism, a unique simple module iff any two totally transcendental matrices of the same order over $D$ are similar. The result applies to the class of existentially closed division algebras and gives a partial solution to a problem posed by Cozzens and Faith.