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Let $R$ be a ring, and let $I, J\subset R$ be two ideals of $R$. Given $n\in \mathbb{N}$, is it true that $(I\cap J)^n=I^n\cap J^n$ ?

I've shown that the inclusion $(I\cap J)^n\subset I^n\cap J^n$ is true, and I'm trying to find a counterexample to the opposite inclusion. I've tried with some rings but I did not find a counterexample.

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    It's certainly true if $I$ and $J$ are comaximal, because then $I\cap J=IJ$. I'm not sure about the general case but I'm thinking about it right now, and I'll update if I find a counterexample – 2017-01-28

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