Multiplicatively closed set

In abstract algebra, a multiplicatively closed set (or multiplicative set) is a subset "S" of a ring "R" such that the following two conditions hold:
In other words, "S" is closed under taking finite products, including the empty product 1.
Equivalently, a multiplicative set is a submonoid of the multiplicative monoid of a ring.
Multiplicative sets are important especially in commutative algebra, where they are used to build localizations of commutative rings.
A subset "S" of a ring "R" is called saturated if it is closed under taking divisors: i.e., whenever a product "xy" is in "S", the elements "x" and "y" are in "S" too.
Examples.
Common examples of multiplicative sets include: