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Let $P,Q$ be ideals such that $P^{2} \subset Q \subset P$ where $\subset$ means proper inclusion and such that $P$ is a prime ideal. Can you please explain why this implies that the ideal $Q$ is never a power of the ideal $P$?

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    this holds even if $P$ is not prime.2011-05-26

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HINT $\ $ divides $\: \Rightarrow\: $ contains:$\rm\ \ \ P^2\ |\ P^{n+2}\ \Rightarrow\ P^2\supset P^{n+2}$