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.
 * $$1 \in S$$,
 * $$xy \in S$$ for all $$x, y \in S$$.

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
Examples of multiplicative sets include:
 * the set-theoretic complement of a prime ideal in a commutative ring;
 * the set {1, x, x2, x3, ...}, where x is an element of a ring;
 * the set of units of a ring;
 * the set of non-zero-divisors in a ring;
 * 1 + I for an ideal I;
 * the Jordan–Pólya numbers, the multiplicative closure of the factorials.

Properties

 * An ideal P of a commutative ring R is prime if and only if its complement R \ P is multiplicatively closed.
 * A subset S is both saturated and multiplicatively closed if and only if S is the complement of a union of prime ideals. In particular, the complement of a prime ideal is both saturated and multiplicatively closed.
 * The intersection of a family of multiplicative sets is a multiplicative set.
 * The intersection of a family of saturated sets is saturated.