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:

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

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.

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.
• 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 $R\setminus S\supseteq P\supseteq I$.
• 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.

See also

• Localization of a ring
• Right denominator set

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.
• Kaplansky, Irving (1974), Commutative rings (Revised ed.), University of Chicago Press, MR 0345945
• Serge Lang, Algebra 3rd ed., Springer, 2002.