Skew power series rings over a prime base ring

TL;DR

This paper studies the structure of skew power series rings over prime rings, focusing on invariance of prime ideals and using valuation theory to address longstanding questions in non-commutative algebra.

Contribution

It introduces new techniques to analyze prime ideal behavior in skew power series rings, especially under conditions where the derivation and automorphism commute.

Findings

Reduction of prime ideal invariance problem to finite-index case in characteristic p

Preliminary results on Iwasawa algebra case with delta as a skew derivation

Nilradical is nearly invariant under (\sigma,\delta) in Noetherian algebras

Abstract

In this paper, we investigate the structure of skew power series rings of the form $S = R[[x;\sigma,\delta]]$, where $R$ is a complete filtered ring and $(\sigma,\delta)$ is a skew derivation respecting the filtration. Our main focus is on the case in which $\sigma\delta = \delta\sigma$, and we aim to use techniques in non-commutative valuation theory to address the long-standing open question: if $P$ is an invariant prime ideal of $R$, is $PS$ a prime ideal of $S$? When $R$ has characteristic $p$, our results reduce this to a finite-index problem. We also give preliminary results in the "Iwasawa algebra" case $\delta = \sigma - \mathrm{id}_R$ in arbitrary characteristic. A key step in our argument will be to show that for a large class of Noetherian algebras, the nilradical is "almost" $(\sigma,\delta)$-invariant in a certain sense.