<?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head> <title>GAP (ResClasses) - Chapter 3: Semilocalizations of the Integers</title> <meta http-equiv="content-type" content="text/html; charset=UTF-8" /> <meta name="generator" content="GAPDoc2HTML" /> <link rel="stylesheet" type="text/css" href="manual.css" /> </head> <body> <div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div> <div class="chlinkprevnexttop"> <a href="chap0.html">Top of Book</a> <a href="chap2.html">Previous Chapter</a> <a href="chap4.html">Next Chapter</a> </div> <p><a id="X80CB0518869B1818" name="X80CB0518869B1818"></a></p> <div class="ChapSects"><a href="chap3.html#X80CB0518869B1818">3. <span class="Heading">Semilocalizations of the Integers</span></a> <div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X7B3B22AC7E6247A4">3.1 <span class="Heading">Entering semilocalizations of the integers</span></a> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X835483D2834D1F60">3.1-1 Z_pi</a></span> </div> <div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X7B76A5528284ACC4">3.2 <span class="Heading">Methods for semilocalizations of the integers</span></a> </div> </div> <h3>3. <span class="Heading">Semilocalizations of the Integers</span></h3> <p>This package implements residue class unions of the semilocalizations Z_(pi) of the ring of integers. It also provides the underlying <strong class="pkg">GAP</strong> implementation of these rings themselves.</p> <p><a id="X7B3B22AC7E6247A4" name="X7B3B22AC7E6247A4"></a></p> <h4>3.1 <span class="Heading">Entering semilocalizations of the integers</span></h4> <p><a id="X835483D2834D1F60" name="X835483D2834D1F60"></a></p> <h5>3.1-1 Z_pi</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> Z_pi</code>( <var class="Arg">pi</var> )</td><td class="tdright">( function )</td></tr></table></div> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> Z_pi</code>( <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div> <p><b>Returns: </b>The ring Z_(pi) or the ring Z_(p), respectively.</p> <p>The returned ring has the property <code class="code">IsZ_pi</code>. The set <var class="Arg">pi</var> of noninvertible primes can be retrieved by the operation <code class="code">NoninvertiblePrimes</code>.</p> <table class="example"> <tr><td><pre> gap> R := Z_pi(2); Z_( 2 ) gap> S := Z_pi([2,5,7]); Z_( 2, 5, 7 ) </pre></td></tr></table> <p><a id="X7B76A5528284ACC4" name="X7B76A5528284ACC4"></a></p> <h4>3.2 <span class="Heading">Methods for semilocalizations of the integers</span></h4> <p>There are methods for the operations <code class="code">in</code>, <code class="code">Intersection</code>, <code class="code">IsSubset</code>, <code class="code">StandardAssociate</code>, <code class="code">Gcd</code>, <code class="code">Lcm</code>, <code class="code">Factors</code> and <code class="code">IsUnit</code> available for semilocalizations of the integers. For the documentation of these operations, see the <strong class="pkg">GAP</strong> reference manual. The standard associate of an element of a ring Z_(pi) is defined by the product of the noninvertible prime factors of its numerator.</p> <table class="example"> <tr><td><pre> gap> 4/7 in R; 3/2 in R; true false gap> Intersection(R,Z_pi([3,11])); IsSubset(R,S); Z_( 2, 3, 11 ) true </pre></td></tr></table> <table class="example"> <tr><td><pre> gap> StandardAssociate(R,-6/7); 2 gap> Gcd(S,90/3,60/17,120/33); 10 gap> Lcm(S,90/3,60/17,120/33); 40 gap> Factors(R,840); [ 105, 2, 2, 2 ] gap> Factors(R,-2/3); [ -1/3, 2 ] gap> IsUnit(S,3/11); true </pre></td></tr></table> <p> </p> <div class="chlinkprevnextbot"> <a href="chap0.html">Top of Book</a> <a href="chap2.html">Previous Chapter</a> <a href="chap4.html">Next Chapter</a> </div> <div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div> <hr /> <p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p> </body> </html>