In
abstract algebra, a multiplicatively closed set (or multiplicative set) is a
subsetS of a
ringR such that the following two conditions hold:[1][2]
,
for all .
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.
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.