# Powers Vs. Powers

## TL;DR

This paper investigates the concept of power stability of ideals in ring extensions, especially between polynomial rings and integral domains, establishing conditions for stability and exploring implications for various classes of rings.

## Contribution

It introduces and studies power stability and ultimate power stability of ideals in ring pairs, providing characterizations and results for polynomial rings over integral domains.

## Key findings

- Power stability of maximal ideals relates to primary decomposition of powers.
- Radical ideals in polynomial rings over Hilbert domains are power stable.
- In Noetherian domains of dimension 1, all radical ideals in polynomial rings are power stable.

## Abstract

Let $ A \subset B$ be rings. An ideal $ J \subset B$ is called power stable in $A$ if $ J^n \cap A = (J\cap A)^n$ for all $ n\geq 1$. Further, $J$ is called ultimately power stable in $A$ if $ J^n \cap A = (J\cap A)^n$ for all $n$ large i.e., $ n \gg 0$. In this note, our focus is to study these concepts for pair of rings $ R \subset R[X]$ where $R$ is an integral domain. Some of the results we prove are: A maximal ideal $\textbf{m}$ in $R[X]$ is power stable in $R$ if and only if $ \wp^t $ is $ \wp-$primary for all $ t \geq 1$ for the prime ideal $\wp = \textbf{m}\cap R$. We use this to prove that for a Hilbert domain $R$, any radical ideal in $R[X]$ which is a finite intersection of G-ideals is power stable in $R$. Further, we prove that if $R$ is a Noetherian integral domain of dimension 1 then any radical ideal in $R[X] $ is power stable in $R$, and if every ideal in $R[X]$ is power stable in $R$ then $R$ is a field. We also show that if $ A \subset B$ are Noetherian rings, and $ I $ is an ideal in $B$ which is ultimately power stable in $A$, then if $ I \cap A = J$ is a radical ideal generated by a regular $A$-sequence, it is power stable. Finally, we give a relationship in power stability and ultimate power stability using the concept of reduction of an ideal (Theorem 3.22).