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:[1][2]

\( 1\in S, \)
\({\displaystyle xy\in S} for all \( x,y\in S. \)

In other words, S is closed under taking finite products, including the empty product 1.[3] 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:

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.

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.[4] 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.