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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.

See also

Localization of a ring
Right denominator set

Notes

Atiyah and Macdonald, p. 36.
Lang, p. 107.
Eisenbud, p. 59.

Kaplansky, p. 2, Theorem 2.

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.

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