On Power Stable Ideals

TL;DR

This paper introduces the concept of power stable ideals in polynomial rings over integral domains, characterizes maximal ideals that are power stable, and explores conditions under which radical ideals are power stable, linking these properties to the nature of the base ring.

Contribution

It defines power stable ideals, characterizes maximal ideals that are power stable, and establishes conditions for radical ideals to be power stable in various types of integral domains.

Findings

Maximal ideals in R[X] are power stable iff P^t is P-primary for all t.

In Hilbert domains, radical ideals that are finite intersections of G-ideals are power stable.

In Noetherian domains of dimension 1, all radical ideals in R[X] are power stable.

Abstract

We define the notion of a power stable ideal in a polynomial ring $ R[X]$ over an integral domain $ R $. It is proved that a maximal ideal $\chi$ $ M $ in $ R[X]$ is power stable if and only if $ P^t $ is $ P$- primary for all $ t\geq 1 $ for the prime ideal $ P = M \cap R $. Using this we prove that for a Hilbert domain $R$ any radical ideal in $R[X]$ which is a finite intersection G-ideals is power stable. Further, we prove that if $ R $ is a Noetherian integral domain of dimension 1 then any radical ideal in $ R[X] $ is power stable. Finally, it is proved that if every ideal in $ R[X]$ is power stable then $ R $ is a field.