Skew Laurent series ring over a Dedekind domain

Abstract

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