Definition:Saturation of Multiplicatively Closed Subset of Ring

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

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

Definition 1
The saturation of $S$ is the smallest saturated multiplicatively closed subset of $A$ containing $S$.

That is, it is the intersection of all saturated 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 a unit of $A$.

Definition 4
The saturation of $S$ is the complement relative to $A$ of the union of prime ideals that are disjoint from $S$:
 * $\ds \map {\operatorname {Sat} } S = A \setminus \bigcup \set {\mathfrak p \in \Spec A: \mathfrak p \cap S = \O}$

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