In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
, a multiplicatively closed set (or multiplicative set) is a
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all 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 unequal, then ''A'' is a proper subset of ...
''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 ...
of the multiplicative
monoid
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.
Monoid ...
of a ring.
Multiplicative sets are important especially in
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
, 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
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s: 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
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
in a commutative ring;
* the set , where ''x'' is an element of a ring;
* the set of
unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (a ...
s 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) \ ...
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
In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module ''R'', so that it consists of fractio ...
*
Right denominator set
In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, ...
Notes
References
*
M. F. Atiyah and
I. G. Macdonald,
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
The University of Chicago Press is the largest and one of the oldest university presses in the United States. It is operated by the University of Chicago and publishes a wide variety of academic titles, including '' The Chicago Manual of Style' ...
, 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