Multiplicatively closed set
From Wikipedia, the free encyclopedia
(Redirected from Multiplicatively closed subset)
- For all x and y in S, the product xy is in S.
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:
- the set-theoretic complement of a prime ideal in a commutative ring;
- the set , where x is a fixed element of the ring;
- the set of units of the ring;
- the set of non-zero-divisors of the ring;
- 1 + I for an ideal I.
- 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. 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.
- Atiyah and Macdonald, p. 36.
- Lang, p. 107.
- Eisenbud, p. 59.
- Kaplansky, p. 2, Theorem 2.
- 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.