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Ideals in the localization of a ring are localizations of ideals.

Lemma 10.9.16. Each ideal $I'$ of $S^{-1}A$ takes the form $S^{-1}I$, where one can take $I$ to be the inverse image of $I'$ in $A$.

Proof. Immediate from Lemma 10.9.15. $\square$

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Suggested slogan: Ideals in the localization of a ring are localizations of ideals.

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