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 $\map {\iota^{\to} } {\mathfrak a} = \map \iota {\mathfrak a} \in J$.
2. For every ideal $\mathfrak b \in J$, its preimage $\map {\iota^{\gets} } {\mathfrak b} = \map {\iota^{-1} } {\mathfrak b} \in I$.
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 statements 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. $\map \iota {\mathfrak p}$ is a prime ideal of $A_S$.

Open embedding of prime spectrum

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

Proof

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Also see

• Correspondence Theorem for Quotient Rings