In abstract algebra, a '''multiplicatively closed set''' (or '''multiplicative set''') is a subset ''S'' of a 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 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 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 {{nowrap|{1, ''x'', ''x''<sup>2</sup>, ''x''<sup>3</sup>, ...}{{null}}}}, where ''x'' is an element of a ring; * the set of units of a ring; * the set of non-zero-divisors in a ring; * {{nowrap|1 + ''I''}} {{Hair space}}for an ideal ''I''; * the Jordan–Pólya numbers, the multiplicative closure of the factorials.

==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 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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