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$ P$-orderings of noncommutative rings


Author: Keith Johnson
Journal: Proc. Amer. Math. Soc. 143 (2015), 3265-3279
MSC (2010): Primary 16S36; Secondary 13F20, 11C08
Published electronically: April 1, 2015
MathSciNet review: 3348770
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Abstract: Let $ K$ be a local field with valuation $ \nu $, $ D$ a division algebra over $ K$ to which $ \nu $ extends, $ R$ the maximal order in $ D$ with respect to $ \nu $ and $ S$ a subset of $ R$. If $ D[x]$ denotes the ring of polynomials over $ D$ with $ x$ a central variable, then the set of integer valued polynomials on $ S$ is $ \mathrm {Int}(S,R)=\{f(x)\in D[x]:f(S)\subseteq R\}$. If $ D$ is commutative, then M. Bhargava showed how to construct a regular $ R$-basis for this set by introducing the idea of a $ P$-ordering of $ S$. We show that this definition can be extended to the noncommutative case in such a way as to construct regular bases there also. We show how to extend methods developed to compute $ P$-orderings in the commutative case and apply them to give a recursive formula for such an ordering for $ D$ the rational quaternions and $ S=R$ the Hurwitz quaternions localized at the prime $ 1+i$.


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Additional Information

Keith Johnson
Affiliation: Department of Mathematics, Dalhousie University, Halifax, Nova Scotia, B3H 4R2, Canada
Email: johnson@mathstat.dal.ca

DOI: https://doi.org/10.1090/S0002-9939-2015-12377-5
Received by editor(s): November 5, 2012
Received by editor(s) in revised form: September 25, 2013
Published electronically: April 1, 2015
Communicated by: Harm Derksen
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.