Extending valuation rings via ultrafilters (Q1911978)

The author continues the study of extendibility of valuation rings in division algebras. He generalizes results from finite-dimensional to locally finite division algebras, i.e. algebras whose finitely generated subalgebras are finite-dimensional. Let \(L\) denote a locally finite division algebra over a division ring \(F\), then the main result is: Given a subdivision algebra \(D\) of \(L\) and a valuation ring \(B\) in \(D\) (i.e. \(x \in B\) or \(x^{-1} \in B\) for any \(x \in D \backslash \{0\})\) then \(B\) can be extended to \(L\) iff \(B \cap F\) can be extended to \(L\) iff the integral closure of \(B \cap F\) in \(L\) is a subring of \(L\). The proof relies on a general sufficient criterion (proved by means of an ultrafilter construction) which yields extendibility to \(L\) even if one only knows it inside arbitrary finite extensions of \(F\).