Krull's intersection theorem says that:

Let $R$ be a Noetherian ring, I an ideal of $R$. Let $N = \cap_{n\geq 1}I^n$. Then $N = IN$.

Here are my questions:

(1) Do there exist a non-noetherian ring $R$ and ideals $I$ and $N$ such that $N = \cap_{n\geq 1}I^n$ but $N \neq IN$?

(2) Consider the question (1) in the case $(R, \frak m)$ is local ring and $I = \frak m$ is a finitely generated.