Filtered skew derivations on simple artinian rings

TL;DR

This paper investigates the structure of skew power series rings over filtered rings, identifying conditions under which the twisting data can be simplified, especially for matrix rings over noncommutative valuation rings.

Contribution

It provides criteria for untwisting skew derivations in filtered rings and analyzes the structure of skew power series rings in specific noncommutative cases.

Findings

Conditions for untwisting skew derivations are established.

Structural analysis of skew power series rings over matrix rings.

Insights into topological obstructions in skew ring constructions.

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 $R[[x; \sigma, \delta]] \cong R[[x'; \sigma]]$, 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.