Definition:Saturation of Ideal by Multiplicatively Closed Subset

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

Generalizations

 * Definition:Saturation of Submodule by Multiplicatively Closed Subset