The ascending chain condition for principal left or right ideals of skew generalized power series rings

TL;DR

This paper investigates the ascending chain condition on principal ideals in skew generalized power series rings, establishing conditions under which these rings are reduced and satisfy ACC, with applications to various types of series rings.

Contribution

It provides new criteria for when skew generalized power series rings satisfy the ascending chain condition on principal ideals, extending to several classical series rings.

Findings

$R[[S, ext{omega}]]$ is right archimedean and reduced under certain conditions.

Power series, Laurent series, and skew series rings satisfy ACC on principal ideals.

Skew Laurent polynomial rings satisfy ACCPL(R) if $R$ is $ ext{alpha}$-rigid and satisfies ACCPL(R).

Abstract

Let $R$ be a ring, $(S,\leq)$ a strictly ordered monoid and $\omega: S\rightarrow End(R)$ a monoid homomorphism. In this paper we study the ascending chain conditions on principal left (resp. right) ideals of the skew generalized power series ring $R[[S,\omega ]]$. Among other results, it is shown that $R[[S,\omega ]]$ is a right archimedean reduced ring if $S$ is an Artinian strictly totally ordered monoid, $R$ is a right archimedean and $S$-rigid ring which satisfies the ACC on annihilators and $\omega_s$ preserves nonunits of $R$ for each $s\in S$. As a consequence we deduce that the power series rings, Laurent series rings, skew power series rings, skew Laurent series rings and generalized power series rings are reduced satisfying the ascending chain condition on principal left (or right) ideals. It is also proved that, the skew Laurent polynomial ring $R[x,x^{-1};\alpha]$ satisfies \emph{ACCPL(R)}, if $R$ is $\alpha$-rigid and satisfies \emph{ACCPL(R)} and the $ACC$ on left(resp. right) annihilators. Examples are provided to illustrate and delimit our results.