A classification of graded extensions in a skew Laurent polynomial ring

Abstract

Let $V$ be a total valuation ring of a division ring $K$ with an automorphism $\sigma$ and let $A={\oplus }_{i\in Z}{A}_i{X}^i$ be a graded extension of $V$ in $K[X,X^{-1};\sigma ]$, the skew Laurent polynomial ring. We classify $A$ by distinguishing four different types based on the properties of $A_1$ and $A_{-1}$. A complete description of $A_i$ for all $i\in \text{Z}$ is given in the case where $A_1$ is a finitely generated left $O_l(A_1)$-ideal.