# 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]}^{[2]}

- .
- For all
*x*and*y*in*S*, the product*xy*is 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 , where
*x*is a fixed element of the ring; - the set of units of the 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

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

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