On Power Stable Ideals

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.