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]

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.


Common examples of multiplicative sets include:


See also


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


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