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Let $(A,\mathfrak{m})$ be a Noetherian local ring. If $I$ is an ideal of $A$, then by (a weak version of) the Artin-Rees lemma, there exists $j \geq 0$ such that for all $i \geq j$, $$\mathfrak{m}^i \cap I \subseteq \mathfrak{m}^{i-j} I.$$

Question: Can we choose $j$ such that the above inclusion holds with $I = \mathrm{Ann}_A(a)$ for all $a \in A$?

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  • $\begingroup$ Do you mean "Can we choose $j$ such that..."? $\endgroup$
    – Kevin Buzzard
    Commented Feb 7, 2017 at 20:25
  • $\begingroup$ Yes, thank you. I fixed the typo in the question. $\endgroup$
    – Arkandias
    Commented Feb 8, 2017 at 12:16

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