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Let $R$ be a commutative noetherian ring with unity, $M$ a finitely generated $R$-module, $I$ an ideal of $R$ such that $\bigcap_{t\ge 1} I^tM=0$ and $M\cong\underset{t}{\varprojlim}M/I^tM$.

Now, let $U\subseteq M$ be a nonzero submodule. From the above conditons, can I claim that there exists $t\in \mathbb N$ s.t. $I^tM\subseteq U?$ (Or it requires some more conditions to the claim hold?)

Thanks.

Q.TL
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1 Answers1

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No, this isn't true in general:

Take $R=\mathbb{Z}_p$ the $p$-adic integers, $M=\mathbb{Z}_p \oplus \mathbb{Z}_p$ and $U=\mathbb{Z}_p \cdot (1,-1) \le \mathbb{Z}_p \oplus \mathbb{Z}_p$.

Then $\hat{M}\cong M\otimes_{\mathbb{Z}_p}\hat{\mathbb Z}_p \cong M\otimes_{\mathbb{Z}_p} \mathbb{Z}_p \cong M$, but each $I^n$ is of the form $I^n=(p^m)$ and hence $I^nM=p^m \mathbb{Z}_p\oplus p^m\mathbb{Z}_p$ is never contained in $U$.

tj_
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