#### Vol. 6, No. 7, 2012

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Free subalgebras of quotient rings of Ore extensions

### Jason P. Bell and Daniel Rogalski

Vol. 6 (2012), No. 7, 1349–1367
##### Abstract

Let $K$ be a field extension of an uncountable base field $k$, let $\sigma$ be a $k$-automorphism of $K$, and let $\delta$ be a $k$-derivation of $K$. We show that if $D$ is one of $K\left(x;\sigma \right)$ or $K\left(x;\delta \right)$, then $D$ either contains a free algebra over $k$ on two generators, or every finitely generated subalgebra of $D$ 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 $k$ is either again PI, or else it contains a free algebra over its center on two variables.

##### Keywords
free algebra, division algebra, Ore extension, skew polynomial ring
##### Mathematical Subject Classification 2010
Primary: 16K40
Secondary: 16S10, 16S36, 16S85