# Localization and Noetherian property

From page 101 in Atiyah-MacDonald:

"Two of the important properties of localization are that it preserves exactness and the Noetherian property...."

I remember proving that it preserves exactness, it's proposition 3.3. on page 39. But what is meant by "Noetherian property"? Thanks.

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## 1 Answer

It means that if $M$ is Noetherian then so is $S^{-1}M$. If you first prove that for any submodule $N'$ of $S^{-1}M$ one has $N' = S^{-1}N$, where $N$ is the inverse image of $N'$ in $M$, then this follows quickly.

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Thank you, I didn't realise it meant that. For rings this is theorem 7.3. on page 80. –  Matt N. Jul 25 '12 at 6:28
@Clark Yep, it's common to prove that (and a few other facts about extension/contraction along $M \to S^{-1}M$) for rings, but it's also true for modules. For this fact I don't see any difference in the proof, really. –  Dylan Moreland Jul 25 '12 at 6:30
Sure, I just wanted to add the comment in case someone else reads this thread and is interested. : ) –  Matt N. Jul 25 '12 at 6:31
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