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