Valuation rings in finite-dimensional division algebras (Q1117304)

By a valuation ring in a division algebra \(D\) the authors understand a subring containing for each \(x\) in \(D\) either \(x\) or \(x^{-1}\). Let \(D\) be a division algebra of dimension \(n^ 2\) over its centre \(K\) and let \(B\) be a valuation ring in \(K\). Then the authors show that in \(D\) there are at most \(n\) valuation rings \(B'\) such that \(B'\cap K=B\), and any two such extensions are conjugate by an element of \(D\). Further the integral closure \(T\) of \(B\) in \(D\) is a subring if and only if \(B\) has extensions to \(D\) and when this is so then \(T\) is the intersection of all these extensions. In the special case when \(K\) is a global field it is shown that only a finite number of valuation rings of \(K\) are extendable to \(D\). These results are illustrated by some examples.