Multiplicatively closed set
From HandWiki
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]
- [math]\displaystyle{ 1 \in S }[/math],
- [math]\displaystyle{ xy \in S }[/math] for all [math]\displaystyle{ x, y \in S }[/math].
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
Examples of multiplicative sets include:
- the set-theoretic complement of a prime ideal in a commutative ring;
- the set {1, x, x2, x3, ...}, where x is an element of a ring;
- the set of units of a ring;
- the set of non-zero-divisors in a ring;
- 1 + I for an ideal I.
- the Jordan–Pólya numbers, the multiplicative closure of the factorials
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
- Localization of a ring
- Right denominator set
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
- Serge Lang, Algebra 3rd ed., Springer, 2002.
Original source: https://en.wikipedia.org/wiki/Multiplicatively closed set.
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