Vol. 6, No. 7, 2012

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
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 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/