Free subalgebras of quotient rings of Ore extensions

Abstract

Let $K$ be a field, let $\ddot{I}ƒ$ be an automorphism of $K$, and let $\hat{I}´$ be a derivation of $K$. We show that if $D$ is one of $K(x;\ddot{I}ƒ)$ or $K(x;\hat{I}´)$, then $D$ either contains a free algebra over its center on two generators, or every finitely generated subalgebra of $D$ satisfies a polynomial identity. As a corollary, we are able to show that the quotient division ring of any iterated Ore extension of an affine domain satisfying a polynomial identity either again satisfies a polynomial identity or it contains a free algebra over its center on two variables.

12 pages; minor changes to statement of main theorem; changes to proofs