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]

  • 1 \in S.
  • 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[edit]

Common examples of multiplicative sets include:

Properties[edit]

  • 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[edit]

Notes[edit]

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

References[edit]