Multiplicatively closed set

<h2 id="examples">Examples</h2>
<p>Examples of multiplicative sets include:
</p>
<ul><li>the <a href="/facts/Set-theoretic_complement/yWUePIYY">set-theoretic complement</a> of a <a href="/facts/Prime_ideal/pmrGf6DA">prime</a> <a href="/facts/Ideal_(ring_theory)/vCvytduX">ideal</a> in a commutative ring;</li>
<li>the set {1, <i>x</i>, <i>x</i>2, <i>x</i>3, ...}, where <i>x</i> is an element of a ring;</li>
<li>the set of <a href="/facts/Unit_(ring_theory)/GJSBu8Y7">units</a> of a ring;</li>
<li>the set of <a href="/facts/Zero_divisor/5GraAU8o">non-zero-divisors</a> in a ring;</li>
<li>1 + <i>I</i>  for an ideal <i>I</i>;</li>
<li>the <a href="/facts/Jordan%E2%80%93P%C3%B3lya_number/blvkWpfR">Jordan–Pólya numbers</a>, the multiplicative closure of the <a href="/facts/Factorial/kkfy1Bwe">factorials</a>.</li></ul>
<h2 id="properties">Properties</h2>
<ul><li>An ideal <i>P</i> of a commutative ring <i>R</i> is prime if and only if its complement <i>R</i> \ <i>P</i> is multiplicatively closed.</li>
<li>A subset <i>S</i> is both saturated and multiplicatively closed if and only if <i>S</i> is the complement of a <a href="/facts/Set-theoretic_union/KVVXKfWl">union</a> of prime ideals.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>  In particular, the complement of a prime ideal is both saturated and multiplicatively closed.</li>
<li>The intersection of a family of multiplicative sets is a multiplicative set.</li>
<li>The intersection of a family of saturated sets is saturated.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Localization_of_a_ring/rjevzBJf">Localization of a ring</a></li>
<li><a href="/facts/Right_denominator_set/FXaaof55">Right denominator set</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/M._F._Atiyah/RQVJ9ATE">M. F. Atiyah</a> and <a href="/facts/I._G._Macdonald/PoaeNbeg">I. G. Macdonald</a>, <i><a href="https://books.google.com/books?id=161SDwAAQBAJ&q=%22multiplicatively+closed%22">Introduction to commutative algebra</a></i>, Addison-Wesley, 1969.</li>
<li><a href="/facts/David_Eisenbud/hw2yEnnx">David Eisenbud</a>, <i><a href="https://books.google.com/books?id=xDwmBQAAQBAJ&q=%22multiplicatively+closed%22">Commutative algebra with a view toward algebraic geometry</a></i>, Springer, 1995.</li>
<li><a href="/facts/Irving_Kaplansky/nRLyp7K9">Kaplansky, Irving</a> (1974), <i>Commutative rings</i> (Revised ed.), <a href="/facts/University_of_Chicago_Press/6se0Ezfb">University of Chicago Press</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0345945">0345945</a></li>
<li><a href="/facts/Serge_Lang/I7MW2rUu">Serge Lang</a>, <i>Algebra</i> 3rd ed., Springer, 2002.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Atiyah and Macdonald, p. 36. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Lang, p. 107. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Eisenbud, p. 59. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Kaplansky, p. 2, Theorem 2. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Multiplicatively closed set open-in-new

Multiplicatively closed set