Let $A$ be Noetherian ring that is an integral domain and let $\frak a$ be a proper ideal. I would like to know if it can happen that $\cap_{1}^{\infty}a^n\ne \cap_{1}^{\infty}{\frak a}^n\ne 0$ and at the same time the sequence of ideals $a^n$ {\frak a}^n$ does not stabilise? What would be a "natural example"?
(this question is motivated by trying to understand Krull intersection theorem).
PS. I was thinking of the following version of Krull intersection theorem):
Theorem. Let a $\frak a$ be be an ideal in a noetherian ring $A$. If $\frak a$ is contained in all maximal ideals of $A$, then $\cap_{1}^{\infty}{\frak a}^n=0$.
