In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
, a multiplicatively closed set (or multiplicative set) is a
subset
In mathematics, a 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 a ...
''S'' of a
ring
(The) Ring(s) 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
Arts, entertainment, and media Film and TV
* ''The Ring'' (franchise), a ...
''R'' such that the following two conditions hold:
*
,
*
for all
.
In other words, ''S'' is
closed under taking finite products, including the
empty product
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplication, multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operat ...
1.
[Eisenbud, p. 59.]
Equivalently, a multiplicative set is a
submonoid
In abstract algebra, 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 .
Monoids are semigroups with identity ...
of the multiplicative
monoid
In abstract algebra, 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 .
Monoids are semigroups with identity ...
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 ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...
, 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 divisibl ...
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 in a commutative ring;
* the set , where ''x'' is an element of a ring;
* the set of
unit
Unit may refer to:
General measurement
* Unit of measurement, a definite magnitude of a physical quantity, defined and adopted by convention or by law
**International System of Units (SI), modern form of the metric system
**English units, histo ...
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 (mathematics), 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 ...
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
Localization or localisation may refer to:
Biology
* Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence
* Localization of sensation, ability to tell what part of the body is aff ...
*
Right denominator set
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 former director of the then Mathematical Sciences Research Institute (MSRI), now k ...
,
Commutative algebra with a view toward algebraic geometry', Springer, 1995.
*
*
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
{{commutative-algebra-stub