Definition:Saturation of Multiplicatively Closed Subset of Ring

Definition
Let $A$ be a commutative ring with unity.

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

Definition 1
The saturation of $S$ is the smallest multiplicatively closed subset of $A$ containing $S$. That is, it is the intersection of all multiplicatively closed subsets containing $S$.

Definition 2
The saturation of $S$ is the set of divisors of elements of $S$.

Definition 3
The saturation of $S$ is the set of elements whose image in the localization $A_S$ is invertible.

Definition 4
The saturation of $S$ is the complement of the union of prime ideals that are disjoint from $S$:
 * $\operatorname{Sat}(S) = A - \displaystyle \bigcup \left\{ \mathfrak p \in \operatorname{Spec} A : \mathfrak p \cap S = \varnothing \right\}$

Also see

 * Equivalence of Definitions of Saturation of Multiplicatively Closed Subset of Ring
 * Definition:Saturated Multiplicatively Closed Subset of Ring
 * Saturation of Multiplicatively Closed Subset of Ring is Closure Operation