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<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>
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<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>
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<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>
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<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>
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