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Multiplicatively closed set
View on WikipediaIn 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.
Examples
[edit]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
[edit]- An ideal P of a commutative ring R is prime if and only if its complement R \ P is multiplicatively closed.
- An ideal P of a commutative ring R that is maximal with respect to being disjoint from a multiplicative set S is a prime ideal (Krull). In fact, if ideal I is disjoint from S, there exists prime ideal P such that .
- 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]References
[edit]- 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, MR 0345945
- Serge Lang, Algebra 3rd ed., Springer, 2002.
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