View source for Multiplicatively closed set

In [[abstract algebra]], a '''multiplicatively closed set''' (or '''multiplicative set''') is a [[subset]] ''S'' of a [[ring (mathematics)|ring]] ''R'' such that the following two conditions hold:<ref>Atiyah and Macdonald, p.&nbsp;36.</ref><ref>Lang, p.&nbsp;107.</ref>
* <math>1 \in S</math>,
* <math>xy \in S</math> for all <math>x, y \in S</math>.
In other words, ''S'' is [[closure (mathematics)|closed]] under taking finite products, including the [[empty product]] 1.<ref>Eisenbud, p.&nbsp;59.</ref>
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 [[localization of a ring|localizations]] of commutative rings.

A subset ''S'' of a ring ''R'' is called '''saturated''' if it is closed under taking [[divisibility (ring theory)|divisor]]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 ideal|prime]] [[Ideal (ring theory)|ideal]] in a commutative ring;
* the set {{nowrap|{1, ''x'', ''x''<sup>2</sup>, ''x''<sup>3</sup>, ...}{{null}}}}, where ''x'' is an element of a ring;
* the set of [[unit (ring theory)|unit]]s of a ring;
* the set of [[zero divisor|non-zero-divisors]] in a ring;
* {{nowrap|1 + ''I''}} {{Hair space}}for an ideal ''I'';
* the [[Jordan–Pólya number]]s, the multiplicative closure of the [[factorial]]s.

==Properties==
* An ideal ''P'' of a commutative ring ''R'' is prime if and only if its complement {{nowrap|''R'' \ ''P''}} is multiplicatively closed. 
* An ideal ''P'' of a commutative ring ''R'' that is maximal with respect to being disjoint from a multiplicative set ''S'' is a prime ideal (Krull).  In fact, if ideal ''I'' is disjoint from ''S'', there exists prime ideal ''P'' such that <math>R\setminus S\supseteq P\supseteq I</math>. 
* A subset ''S'' is both saturated and multiplicatively closed if and only if ''S'' is the complement of a [[set-theoretic union|union]] of prime ideals.<ref name="Kap2">Kaplansky, p.&nbsp;2, Theorem&nbsp;2.</ref>  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==
{{reflist}}

== 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.


[[Category:Commutative algebra]]


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