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<head><title>idealizer -- compute Hom(I,I) as a quotient ring</title>
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<div><h1>idealizer -- compute Hom(I,I) as a quotient ring</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>(F,G) = idealizer(I,f)</tt></div>
</dd></dl>
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</li>
<li><div class="single">Inputs:<ul><li><span><tt>I</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, in a domain <i>R</i></span></li>
<li><span><tt>f</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span>, an element of the ideal <i>I</i></span></li>
<li><span><tt>Variable</tt>, <span>a <a href="../../Macaulay2Doc/html/___Symbol.html">symbol</a></span></span></li>
<li><span><tt>Index</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span></span></li>
</ul>
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</li>
<li><div class="single">Outputs:<ul><li><span><tt>F</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Map.html">ring map</a></span>, The inclusion map from <i>R</i> into <i>S = Hom<sub>R</sub>(I,I)</i></span></li>
<li><span><tt>G</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Map.html">ring map</a></span>, <i>frac S →frac R</i>, giving the fractions corresponding to each generator of <i>S</i>.</span></li>
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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="_idealizer_lp..._cm_sp__Index_sp_eq_gt_sp..._rp.html">Index => ...</a>, -- Sets the starting index on the new variables used to build the endomorphism ring Hom(J,J). If the program idealizer is used independently, the user will generally want to use the default value of 0. However, when used as part of the integralClosure computation the number needs to start higher depending on the level of recursion involved. </span></li>
<li><span><tt>Strategy => ...</tt> (missing documentation<!-- tag: idealizer(..., Strategy => ...) -->), </span></li>
<li><span><a href="_idealizer_lp..._cm_sp__Variable_sp_eq_gt_sp..._rp.html">Variable => ...</a>, -- Sets the name of the indexed variables introduced in computing the endomorphism ring Hom(J,J).</span></li>
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<div class="single"><h2>Description</h2>
<div><div>This is a key subroutine used in the computation of integral closures.</div>
<table class="examples"><tr><td><pre>i1 : R = QQ[x,y]/(y^3-x^7)
o1 = R
o1 : QuotientRing</pre>
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<tr><td><pre>i2 : I = ideal(x^2,y^2)
2 2
o2 = ideal (x , y )
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : (F,G) = idealizer(I,x^2);</pre>
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<tr><td><pre>i4 : target F
QQ[w , x, y]
0,0
o4 = -------------------------------------
5 2 2 2 3
(w y - x , w x - y , w - x y)
0,0 0,0 0,0
o4 : QuotientRing</pre>
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<tr><td><pre>i5 : first entries G.matrix
2
y
o5 = {--, x, y}
2
x
o5 : List</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_ring__From__Fractions.html" title="find presentation for f.g. ring">ringFromFractions</a> -- find presentation for f.g. ring</span></li>
<li><span><a href="_integral__Closure.html" title="integral closure of an ideal or a domain">integralClosure</a> -- integral closure of an ideal or a domain</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>idealizer</tt> :</h2>
<ul><li>idealizer(Ideal,RingElement)</li>
</ul>
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