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

Abstract

Let K be a division ring with a σ-derivation δ, where σ is an endomor-phism of K and K(X;σ, δ) be the quotient division ring of the Ore extension K[X;σ, δ] over K in an indeterminate X. First, we describe non-commutative valuation rings of K(X;σ, δ) which contain K[X;σ, δ]. Suppose that (σ, δ) is compatible with V, where V is a total valuation ring of K, then R⁽¹⁾=V[X;σ, δ]_{J(V)[X;σ, δ]}, the localization of V[X;σ, δ] at J(V)[X;σ, δ], is a total valuation ring of K(X;σ, δ). Applying the description above, then, second, we describe non-commutative valuation rings B of K(X;σ, δ) such that B∩ K=V, X∈ B and B⊂eq 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.