More localization

The localization is an exact functor.

We have $S^{-1}A\otimes_A M\cong S^{-1}M$ as A-modules and $S^{-1}A$ -modules.

Local properties: an A-module M is zero if and only if its localizations at prime ideals are, if and only if the localizations at the maximal ideals are.

In $S^{-1}A$ all ideals are extension of ideals of A. An ideal I of A extends to the whole ring if and only if it meets S. Prime ideals of the localization are in 1-1 correspondence to prime ideals of A which do not meet S. The nilradical of the localization is the localization of the nilradical.

References: Atiyah-Macdonald chap 3.

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