Definition:Saturation of Ideal by Multiplicatively Closed Subset
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Definition
Let $A$ be a commutative ring with unity.
Let $\mathfrak a \subseteq A$ be an ideal.
Let $S \subseteq A$ be a multiplicatively closed subset.
Definition 1
The saturation of $\mathfrak a$ by $S$ is the ideal:
- $\{ a \in A : \exists s \in S : as \in \mathfrak a\}$
Definition 2
Let $A \overset \iota \to A_S$ be the localization of $A$ at $S$.
The saturation of $\mathfrak a$ is the preimage of its image under the ring homomorphism $\iota : A \to A_S$:
- $\iota^{-1}(\iota(\mathfrak a))$
Also see
- Saturation of Ideal by Multiplicatively Closed Subset is Ideal
- Correspondence Theorem for Localizations of Rings