On Reduced archimedean skew power series rings

TL;DR

This paper investigates conditions under which skew power series rings over Archimedean reduced rings retain the Archimedean and reduced properties, extending classical results to non-commutative settings.

Contribution

It generalizes the preservation of Archimedean and reduced properties to skew power series rings with rigid automorphisms under ACC on annihilators.

Findings

$R[[x]]$ is Archimedean and reduced if $R$ is, under ACC.

Skew power series rings $R[[x;\alpha]]$ are Archimedean and reduced with a rigid automorphism.

Examples justify the necessity of the assumptions.

Abstract

In this paper, we prove that if $R$ is an Archimedean reduced ring and satisfy ACC on annihilators, then $R[[x]]$ is also an Archimedean reduced ring. More generally we prove that if $R$ is a right Archimedean ring satisfying the \emph{ACC} on annihilators and $\alpha$ is a rigid automorphism of $R$, then the skew power series ring $R[[x;\alpha ]]$ is right Archimedean reduced ring. We also provide some examples to justify the assumptions we made to obtain the required result.