# 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 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 $S$:

$\map {\operatorname {Sat} } S = A \setminus \displaystyle \bigcup \set {\mathfrak p \in \operatorname{Spec} A: \mathfrak p \cap S = \O}$