A classification of graded extensions in a skew Laurent polynomial ring, II

Abstract

Let $V$ be a total valuation ring of a division ring $K$ with an automorphism $\sigma$ and let $A=\oplus_{i\in \mbi{Z}} A_{i} X^{i}$ be a graded extension of $V$ in $K[X,X^{-1};\sigma]$, the skew Laurent polynomial ring. We classify $A$ by distinguishing three different types based on the properties of $A_{1}$ and $A_{-1}$, and a complete description of $A_{i}$ for all $i\in \mbi{Z}$ is given in the case where $A_{1}$ is not a finitely generated left $O_{l}(A_{1})$-ideal.