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

Abstract

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 $R^{\left(1\right)}=V[X;\sigma ,\delta {]}_{J\left(V\right)[X;\sigma ,\delta ]}$, the localization of $V[X;\sigma ,\delta ]$ at $J\left(V\right)[X;\sigma ,\delta ]$, 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\cap K=V,$$X\in B$ and $B\subseteq R^{\left(1\right)}$, which is the aim of this paper. In the end of each section we give several examples to display some of the various phenomena.