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:Atiyah and Macdonald, p. 36.Lang, p. 107. , for all . In other words, S is closed under taking finite products, including the empty product 1.Eisenbud, p. 59. 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 (Krull). 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.Kaplansky, p. 2, Theorem 2. 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.
See also Localization of a ring Right denominator set
Notes
References M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley, 1969. David Eisenbud, Commutative algebra with a view toward algebraic geometry, Springer, 1995.
Serge Lang, Algebra 3rd ed., Springer, 2002.