On Power Stable Ideals

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.