Extending Real (C1;C2)-holder Valuation T0 Skew Polynomial Ring

The aim of this paper is as to study real (C1,C2)- Holder valuations on skew polynomials rings. Let D be a division ring, T be a variable over D,σ an endomorphism of D, δ a σ-derivation of D and R = D[T; σ; δ] the left skew polynomial ring over D. We show the set (HV alv(R),) of σ-compatible real Holder valuations which extend as to R a fixed proper real Holder valuation ⊆ on D, has a natural structure of parameterized complete non-metric, where is the partial order given by μ ≤ , if and only if μ(f) ≤ (f), for all f ∈ R and μ, ∈ HV alv(R).