Extending real valuations to skew polynomial rings

Abstract

Let $D$ be a division ring, $T$ be a variable over $D$, $\sigma$ be an endomorphism of $D$, $\delta$ be a $\sigma$-derivation on $D$ and $R=D[T; \sigma , \delta]$ the left skew polynomial ring over $D$. We show that the partially ordered set $(Val_\nu(R),\preceq)$ of $\sigma$-compatible real valuations on $R$ extending a fixed proper real valuation $\nu$ on $D$ has a natural structure of complete parameterized non-metric tree.