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<div class="ChapSects"><a href="chap3_mj.html#X80CB0518869B1818">3 <span class="Heading">Semilocalizations of the Integers</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7B3B22AC7E6247A4">3.1 <span class="Heading">Entering semilocalizations of the integers</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7861432E7F221610">3.1-1 Z_pi</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7E75C4217DCA45D0">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 <span class="SimpleMath">\(ℤ_{(\pi)}\)</span> 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="X7861432E7F221610" name="X7861432E7F221610"></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>Returns: the ring <span class="SimpleMath">\(ℤ_{(\pi)}\)</span> or the ring <span class="SimpleMath">\(ℤ_{(p)}\)</span>, respectively.</p>

<p>The returned ring has the property <code class="code">IsZ_pi</code>. The set <var class="Arg">pi</var> of non-invertible primes can be retrieved by the operation <code class="code">NoninvertiblePrimes</code>.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">R := Z_pi(2);</span>
Z_( 2 )
<span class="GAPprompt">gap></span> <span class="GAPinput">S := Z_pi([2,5,7]);</span>
Z_( 2, 5, 7 )

</pre></div>

<p><a id="X7E75C4217DCA45D0" name="X7E75C4217DCA45D0"></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 <span class="SimpleMath">\(ℤ_{(\pi)}\)</span> is defined by the product of the non-invertible prime factors of its numerator.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">4/7 in R; 3/2 in R;</span>
true
false
<span class="GAPprompt">gap></span> <span class="GAPinput">Intersection(R,Z_pi([3,11])); IsSubset(R,S);</span>
Z_( 2, 3, 11 )
true

</pre></div>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">StandardAssociate(R,-6/7);</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">Gcd(S,90/3,60/17,120/33);</span>
10
<span class="GAPprompt">gap></span> <span class="GAPinput">Lcm(S,90/3,60/17,120/33);</span>
40
<span class="GAPprompt">gap></span> <span class="GAPinput">Factors(R,840);</span>
[ 105, 2, 2, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Factors(R,-2/3);</span>
[ -1/3, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsUnit(S,3/11);</span>
true

</pre></div>

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