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

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.

free algebra, division algebra, Ore extension, skew polynomial ring
Mathematical Subject Classification 2010
Primary: 16K40
Secondary: 16S10, 16S36, 16S85
Received: 10 March 2011
Revised: 6 January 2012
Accepted: 7 February 2012
Published: 4 December 2012
Jason P. Bell
Department of Mathematics
Simon Fraser University
8888 University Drive
Burnaby, BC  V5A 1S6
Daniel Rogalski
Department of Mathematics
University of California, San Diego
La Jolla, CA 92093-0112
United States