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Let $N,P$ be submodules of an $R$-module $M$ and let $S$ be a multiplicative subset of $R$. I think I proved $S^{-1}(N \cap P) = S^{-1}N \cap S^{-1} P$ but since my proof is not the same as the one given in Atiyah-MacDonald on page 39 I suspect there is something wrong with it. Can you tell me please what's wrong here:

Claim: $S^{-1}(N \cap P) = S^{-1}N \cap S^{-1} P$

Proof:

$\frac{m}{s} \in S^{-1}N \cap S^{-1} P \iff$ $\frac{m}{s} \in S^{-1}N$ and $\frac{m}{s} \in S^{-1}P \iff m \in N$ and $m \in P \iff m \in N \cap P \iff \frac{m}{s} \in S^{-1}(N \cap P)$.

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    Looks like they were posted approxi$m$ately simultaneously.2012-06-13

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In your notation, it doesn't have to be the case that $m \in N$. There just needs to be an $s' \in S$ such that $s'm \in N$. For example, take $R = M = \mathbf Z$, $N = 6\mathbf Z$, and $S = \{1, 2, 2^2, \ldots\}$. But having made this change, I think you can complete your proof.

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    If $sm \in N$ and $s^\prime m \in P$ then $ss^\prime m \in N \cap P$ since $N,P$ are submodules and hence closed under the action of $R$. : ) I'm glad I asked this question. Thank you!2012-06-13