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:Atiyah and Macdonald, p. 36.Lang, p. 107.
- ,
- for all .
In other words, S is closed under taking finite products, including the empty product 1.Eisenbud, p. 59.
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 {{nowrap|{1, x, x2, x3, ...}{{null}}}}, where x is an element of a ring;
- the set of units of a ring;
- the set of non-zero-divisors in a ring;
- {{nowrap|1 + I}} {{Hair space}}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 {{nowrap|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.Kaplansky, p. 2, Theorem 2. 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
Notes
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References
- M. F. Atiyah and I. G. Macdonald, [https://books.google.com/books?id=161SDwAAQBAJ&q=%22multiplicatively+closed%22 Introduction to commutative algebra], Addison-Wesley, 1969.
- David Eisenbud, [https://books.google.com/books?id=xDwmBQAAQBAJ&q=%22multiplicatively+closed%22 Commutative algebra with a view toward algebraic geometry], Springer, 1995.
- {{Citation | last1=Kaplansky | first1=Irving | author1-link=Irving Kaplansky | title=Commutative rings | publisher=University of Chicago Press | edition=Revised |mr=0345945 | year=1974}}
- Serge Lang, Algebra 3rd ed., Springer, 2002.
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