In
abstract algebra, a multiplicatively closed set (or multiplicative set) is a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
''S'' of a
ring ''R'' such that the following two conditions hold:
*
,
*
for all
.
In other words, ''S'' is
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
under taking finite products, including the
empty product 1.
[Eisenbud, p. 59.]
Equivalently, a multiplicative set is a
submonoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids a ...
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 , where ''x'' is an element of a ring;
* the set of
units of a ring;
* the set of
non-zero-divisors in a ring;
* for an ideal ''I''.
* the
Jordan–Pólya numbers, the multiplicative closure of the
factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) \t ...
s
Properties
* An ideal ''P'' of a commutative ring ''R'' is prime if and only if its complement 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
*
Localization of a ring
*
Right denominator set
Notes
References
*
M. F. Atiyah and
I. G. Macdonald
Ian Grant Macdonald (born 11 October 1928 in London, England) is a British mathematician known for his contributions to symmetric functions, special functions, Lie algebra theory and other aspects of algebra, algebraic combinatorics, and combi ...
,
Introduction to commutative algebra', Addison-Wesley, 1969.
*
David Eisenbud
David Eisenbud (born 8 April 1947 in New York City) is an American mathematician. He is a professor of mathematics at the University of California, Berkeley and Director of the Mathematical Sciences Research Institute (MSRI); he previously serve ...
,
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
Serge Lang (; May 19, 1927 – September 12, 2005) was a French-American mathematician and activist who taught at Yale University for most of his career. He is known for his work in number theory and for his mathematics textbooks, including the i ...
, ''Algebra'' 3rd ed., Springer, 2002.
Commutative algebra