Powers Vs. Powers

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).