Bounded skew power series rings for inner $\sigma$-derivations

TL;DR

This paper introduces and analyzes bounded skew power series rings over certain prime rings, establishing conditions for their primeness and simplicity, especially when involving inner $\sigma$-derivations.

Contribution

It provides new criteria for the well-definedness, primeness, and simplicity of bounded skew power series rings with inner $\sigma$-derivations over complete filtered Noetherian prime rings.

Findings

Under specific conditions, the skew power series ring is often prime.

The ring can be simple if certain restrictions on $\delta$ are met.

The results apply to rings with characteristic $p$ and inner $\sigma$-derivations.

Abstract

We define and explore the bounded skew power series ring $R^+[[x;\sigma,\delta]]$ defined over a complete, filtered, Noetherian prime ring $R$ with a commuting skew derivation $(\sigma,\delta)$. We establish precise criteria for when this ring is well-defined, and for an appropriate completion $Q$ of $Q(R)$, we prove that if $Q$ has characteristic $p$, $\delta$ is an inner $\sigma$-derivation and no positive power of $\sigma$ is inner as an automorphism of $Q$, then $Q^+[[x;\sigma,\delta]]$ is often prime, and even simple under certain mild restrictions on $\delta$. It follows from this result that $R^+[[x;\sigma,\delta]]$ is itself prime.