Valuation rings in ore extensions

Extensions of valuation rings V of a skew field K are considered in the ske w field F = K(x, sigma) for sigma a monomorphism of K. At least two such ex tensions exist if a is an automorphism, but no extension may exist if a 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 a is co mpatible with V. (C) 2001 Academic Press.