Multiplicatively closed set

• 1 ∈ S,
• xy ∈ S for all x and y in S.

Examples

• the set-theoretic complement of a prime ideal in a commutative ring;
• the set, where x is an element of a ring;
• the set of units of a ring;
• the set of non-zero-divisors in a ring;
• for an ideal I.

  Properties

• An ideal P of a commutative ring R is prime if and only if its complement 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.