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

Generalizations