Valuation Rings in Ore Extensions

AbstractExtensions of valuation rings V of a skew field K are considered in the skew field F=K(x,σ) for σ a monomorphism of K. At least two such extensions exist if σ is an automorphism, but no extension may exist if σ is a monomorphism only. There exist extensions R(a) of V in F with xR(a)=aR(a) for every non-zero a in K if and only if V is invariant and σ is compatible with V