Filtered skew derivations on simple artinian rings

Abstract

Given a complete, positively filtered ring $(R,f)$ and a compatible skew derivation $(\sigma ,\delta )$, we may construct its skew power series ring $R[[x;\sigma ,\delta ]]$. Due to topological obstructions, even if $\delta$ is an \emph{inner} $\sigma$-derivation, in general we cannot ``untwist" it, i.e. reparametrise to find a filtered isomorphism $R[[x;\sigma ,\delta ]]\cong R[[x^{\prime };\sigma ]]$, as might be expected from the theory of skew polynomial rings; similarly when $\sigma$ is an inner automorphism. We find general conditions under which it is possible to untwist the multiplication data, and use this to analyse the structure of $R[[x;\sigma ,\delta ]]$ in the simplest case when $R$ is a matrix ring over a (noncommutative) noetherian discrete valuation ring.

17 pages