# Multiplicatively closed set

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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]

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

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:

## Properties

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

## Notes

1. ^ Atiyah and Macdonald, p. 36.
2. ^ Lang, p. 107.
3. ^ Eisenbud, p. 59.
4. ^ 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.