Let R be a local ring, I an ideal, M a finitely generated module and $N=\cap _nI^nM$. Then the Krull intersection theorem states that $N=IN$. Now if R is a local ring of characteristic $p>0$, for each $e\geq 0$ let $I^e$ denote the ideal generated by the $p^e$-th power of the elements of $I$. let $N=\cap _eI^eM$. Is it ture that $N=I^0N$?
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First Krull's theorem is for Noetherian (not necessarily local) rings. Let $n\ge 1$. If $I$ is generated by $r$ elements $x_1, \dots, x_r$, then the usual $n$-power $I^n$ of $I$ is contained in your $I^e$ if $n/r \ge p^e$ (any element of $I^n$ is a combination of $x_1^{a_1}...x_r^{a_r}$ with $a_1+\dots + a_r=n$, so $\max_i{a_r} \ge n/r$). Therefore your $N$ is equal to the usual $\cap_n I^nM$ and the answer to your question is yes. |
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