Non-commutative valuation rings of K (X; σ, δ) over a division ring K

Let K be a division ring with a sigma-derivation delta, where sigma is an endomorphism of K and K(X;sigma,delta) be the quotient division ring of the Ore extension K[X;sigma,delta] over K in an indeterminate X. First, we describe non-commutative valuation rings of K(X;sigma,delta) which contain K[X;sigma,delta]. Suppose that (sigma,delta) is compatible with V, where V is a total valuation ring of K, then R-(1) = V[X;sigma,delta](J(V)[X;sigma,delta]), the localization of V[X;sigma,delta] at J(V)[X;sigma,delta], is a total valuation ring of K(X;sigma,delta). Applying the description above, then, second, we describe non-commutative valuation rings B of K(X;sigma,delta) such that B boolean AND K = V, X is an element of B and B subset of or equal to R-(1), which is the aim of this paper. In the end of each section we give several examples to display some of the various phenomena.