Multiplicatively closed set
In abstract algebra, a multiplicatively closed set is a subset S of a ring R such that the following two conditions hold:
- ,
- for all.
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
Examples of multiplicative sets include:- 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;
- 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 is multiplicatively closed.
- An ideal P of a commutative ring R that is maximal with respect to being disjoint from a multiplicative set S is a prime ideal. In fact, if ideal I is disjoint from S, there exists prime ideal P such that.
- 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.