power of radical of ideal

Question

Let $R$ be a commutative ring. Let $I$ be an ideal of $R$. Let radical of $I$, $J＝√I$. Then, I'm wondering $J^n⊂I$ for some positive integer $n$?

I don't see whether this is true or not. There may be some counterexamples.

References are also appreciated.

P.S 　Thank you for counterexample. If $R$ is Noether, the statement is correct? Any reference is also appreciated.

Answer 1

Consider $R=k[x_1,x_2,\dots]$ for a field $k$ (the polynomial ring in countably infinite many variables.) Take $I=(x_1,x_2^2,x_3^3,\dots)$. Then $J=\sqrt{I}=(x_1,x_2,x_3,\dots)$, but $J^n$ is not contained in $I$ for any $n$.