# Skew power series rings with automorphisms of finite inner order

## TL;DR

This paper studies the algebraic structure of skew power series rings over certain filtered algebras, extending previous results to cases where automorphisms have finite inner order, with implications for prime ideal classification.

## Contribution

It extends the understanding of skew power series rings by analyzing cases with automorphisms of finite inner order, solving a key open problem and impacting Iwasawa algebra theory.

## Key findings

- The skew power series ring is often simple when automorphisms have finite inner order.
- The ring is always prime in relevant cases.
- Extension of simplicity results to finite inner order automorphisms.

## Abstract

We investigate the algebraic properties of the bounded skew power series ring $Q^+[[x;\sigma,\delta]]$ over a (complete, simple) \emph{standard} filtered artinian algebra $Q$ of positive characteristic. Here we are assuming that $(\sigma,\delta)$ is a commuting skew derivation of $Q$, where $\delta$ is inner, satisfying the appropriate compatibility conditions. In a previous work of the authors, it was proved that $Q^+[[x;\sigma,\delta]]$ is a simple ring whenever $\sigma$ has infinite inner order. We now extend this result to the case when $\sigma$ has finite inner order, proving that this ring is often simple, and always prime in cases of interest. This solves an important special case of an open question of Letzter, and yields important consequences for the classification of prime ideals in Iwasawa algebras of solvable groups.