Free subalgebras of quotient rings of Ore extensions

Abstract

Let [math] be a field extension of an uncountable base field [math] , let [math] be a [math] -automorphism of [math] , and let [math] be a [math] -derivation of [math] . We show that if [math] is one of [math] or [math] , then [math] either contains a free algebra over [math] on two generators, or every finitely generated subalgebra of [math] satisfies a polynomial identity. As a corollary, we show that the quotient division ring of any iterated Ore extension of an affine PI domain over [math] is either again PI, or else it contains a free algebra over its center on two variables.