$P$-orderings of noncommutative rings

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 $1+i$.