Skew Laurent Series Ring Over a Dedekind Domain

We show that the formal skew Laurent series ring $R=D(\text{\hspace{0.17em}⁣}(x;\sigma )\text{\hspace{0.17em}⁣})$ over a commutative Dedekind domain $D$ with an automorphism $\sigma$ is a noncommutative Dedekind domain. If $\sigma$ acts trivially on the ideal class group of $D$, then $K_0(R)$, the Grothendieck group of $R$, is isomorphic to $K_0(D)$. Furthermore, we determine the Krull dimension, the global dimension, the general linear rank, and the stable rank of $R$.