Correspondence Theorem for Localizations of Rings

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Theorem

Let $A$ be a commutative ring with unity.

Let $S\subseteq A$ be a multiplicatively closed subset.

Let $A \overset \iota \to A_S$ be the localization at $S$.

Let $I$ be the set of saturated ideals of $A$ by $S$.

Let $J$ be the set of ideals of $A_S$.


Bijection

The direct image mapping $\iota^\to$ and the inverse image mapping $\iota^\gets$ induce reverse bijections between $I$ and $J$, specifically:

  1. For every ideal $\mathfrak a \in I$, its image $\iota^{\to}(\mathfrak a) = \iota(\mathfrak a) \in J$.
  2. For every ideal $\mathfrak b \in J$, its preimage $\iota^{\gets}(\mathfrak b) = \iota^{-1}(\mathfrak b) \in I$.
  3. The restrictions $\iota^\to : I \to J$ and $\iota^\gets : J \to I$ are reverse bijections.


Prime ideals

Let $\mathfrak p \subseteq A$ be an ideal.


The following are equivalent:

  1. $\mathfrak p$ is a prime ideal of $A$ disjoint from $S$.
  2. $\mathfrak p$ is a prime ideal of $A$ saturated by $S$.
  3. $\iota(\mathfrak p)$ is a prime ideal of $A_S$.


Open embedding of prime spectrum

The induced map on spectra $\operatorname{Spec} \iota : \operatorname{Spec} A_S \to \operatorname{Spec} A$ is a topological open embedding.


Also see