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<head><title>localize(Ideal,Ideal) -- localize an ideal at a prime ideal</title>
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<div><a href="index.html" title="">PrimaryDecomposition</a> > <a href="_localize_lp__Ideal_cm__Ideal_rp.html" title="localize an ideal at a prime ideal">localize(Ideal,Ideal)</a></div>
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<div><h1>localize(Ideal,Ideal) -- localize an ideal at a prime ideal</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>localize(I,P)</tt></div>
</dd></dl>
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</li>
<li><span>Function: <a href="_localize_lp__Ideal_cm__Ideal_rp.html" title="localize an ideal at a prime ideal">localize</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>I</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, an ideal in a (quotient of a) polynomial ring <tt>R</tt></span></li>
<li><span><tt>P</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, a prime ideal in the same ring</span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, the extension contraction ideal I R_P intersect R.</span></li>
</ul>
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</li>
<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_localize_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html">Strategy => ...</a>, </span></li>
</ul>
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</li>
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<div class="single"><h2>Description</h2>
<div>The result is the ideal obtained by first extending to the localized ring and then contracting back to the original ring.<table class="examples"><tr><td><pre>i1 : R = ZZ/(101)[x,y];</pre>
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<tr><td><pre>i2 : I = ideal (x^2,x*y);
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : P1 = ideal (x);
o3 : Ideal of R</pre>
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<tr><td><pre>i4 : localize(I,P1)
o4 = ideal(x)
o4 : Ideal of R</pre>
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<tr><td><pre>i5 : P2 = ideal (x,y);
o5 : Ideal of R</pre>
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<tr><td><pre>i6 : localize(I,P2)
2
o6 = ideal (x , x*y)
o6 : Ideal of R</pre>
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<table class="examples"><tr><td><pre>i7 : R = ZZ/31991[x,y,z];</pre>
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<tr><td><pre>i8 : I = ideal(x^2,x*z,y*z);
o8 : Ideal of R</pre>
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<tr><td><pre>i9 : P1 = ideal(x,y);
o9 : Ideal of R</pre>
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<tr><td><pre>i10 : localize(I,P1)
o10 = ideal (y, x)
o10 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i11 : P2 = ideal(x,z);
o11 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i12 : localize(I,P2)
2
o12 = ideal (z, x )
o12 : Ideal of R</pre>
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<b>Author and maintainer: </b>C. Yackel, cyackel@math.indiana.edu. Last modified June 2000.</div>
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<div class="single"><h2>Caveat</h2>
<div>The ideal P is not checked to be prime.</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_primary__Decomposition.html" title="irredundant primary decomposition of an ideal">primaryDecomposition(Ideal)</a> -- irredundant primary decomposition of an ideal</span></li>
<li><span><a href="../../Macaulay2Doc/html/_radical.html" title="the radical of an ideal">radical</a> -- the radical of an ideal</span></li>
<li><span><a href="../../Macaulay2Doc/html/_minimal__Primes.html" title="minimal associated primes of an ideal">minimalPrimes</a> -- minimal associated primes of an ideal</span></li>
<li><span><a href="../../Macaulay2Doc/html/_top__Components.html" title="compute top dimensional component">topComponents</a> -- compute top dimensional component</span></li>
<li><span><a href="../../Macaulay2Doc/html/_remove__Lowest__Dimension.html" title="remove components of lowest dimension">removeLowestDimension</a> -- remove components of lowest dimension</span></li>
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