Real Valuations on Skew Polynomial Rings

Let D be a division ring, T be a variable over D, σ be an endomorphism of D, δ be a σ-derivation on D and R?=?DT; σ, δ] the left skew polynomial ring over D. We show that the set \((Val_\nu(R),\preceq)\) of σ-compatible real valuations which extend to R a fixed proper real valuation ν on D has a natural structure of parameterized complete non-metric tree, where \(\preceq \) is the partial order given by \(\mu \preceq \widetilde{\mu}\) if and only if \(\mu (f)\leq \widetilde{\mu}(f)\) for all f?∈?R and \(\mu, \widetilde{\mu} \in Val_\nu (R)\) . Furthermore and as a consequence, we also prove a criterion of irreducibility for left skew polynomials that includes as a particular case an Eisenstein valuation criterion which generalizes a similar one of Churchill and Zhang (J Algebra 322:3797–3822, 2009).