Free subalgebras of quotient rings of Ore extensions

Bell, Jason; Rogalski, Daniel

doi:10.2140/ant.2012.6.1349

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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

DOI: 10.2140/ant.2012.6.1349

Abstract

Let $K$ be a field extension of an uncountable base field $k$ , let $σ$ be a $k$ -automorphism of $K$ , and let $δ$ be a $k$ -derivation of $K$ . We show that if $D$ is one of $K (x; σ)$ or $K (x; δ)$ , 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

Milestones

Received: 10 March 2011

Revised: 6 January 2012

Accepted: 7 February 2012

Published: 4 December 2012

Authors

Jason P. Bell
Department of Mathematics Simon Fraser University 8888 University Drive Burnaby, BC V5A 1S6 Canada
http://www.math.sfu.ca/~jpb/
Daniel Rogalski
Department of Mathematics University of California, San Diego La Jolla, CA 92093-0112 United States
http://www.math.ucsd.edu/~drogalsk/

Vol. 6, No. 7, 2012