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 term ' ...
, a multiplicatively closed set (or multiplicative set) is a
subset ''S'' of a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
''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.
Monoids a ...
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. Prominent ...
, 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
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
units of a ring;
* the set of
non-zero-divisors in a ring;
* for an ideal ''I''.
* the
Jordan–Pólya number
In mathematics, the Jordan–Pólya numbers are the numbers that can be obtained by multiplying together one or more factorials, not required to be distinct from each other. For instance, 480 is a Jordan–Pólya number because Every tree has a nu ...
s, 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
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''U ...
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
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,
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