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<head><title>ideal(Ring) -- returns the defining ideal</title>
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<div><h1>ideal(Ring) -- returns the defining 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>ideal R</tt></div>
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<li><span>Function: <a href="_ideal.html" title="make an ideal">ideal</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>R</tt>, <span>a <a href="___Ring.html">ring</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>an <a href="___Ideal.html">ideal</a></span>, which is the defining ideal of <tt>R</tt></span></li>
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<div class="single"><h2>Description</h2>
<div>A <a href="___Quotient__Ring.html">quotient ring</a> is a the quotient of its <a href="_ambient.html" title="ambient free module of a subquotient, or ambient ring">ambient</a> <a href="___Ring.html">ring</a> by its defining ideal. Other rings have no ambient ring, and the defining ideal is its zero ideal.<table class="examples"><tr><td><pre>i1 : S = ZZ/2[x,y,z];</pre>
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<tr><td><pre>i2 : ideal S
o2 = ideal ()
o2 : Ideal of S</pre>
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<tr><td><pre>i3 : R = S/(y^2-x*z,x^2*y-z^2)
o3 = R
o3 : QuotientRing</pre>
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<tr><td><pre>i4 : ideal R
2 2 2
o4 = ideal (y + x*z, x y + z )
o4 : Ideal of S</pre>
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<tr><td><pre>i5 : T = R/(x^3-y*z)
o5 = T
o5 : QuotientRing</pre>
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<tr><td><pre>i6 : ideal T
3
o6 = ideal(x + y*z)
o6 : Ideal of R</pre>
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<tr><td><pre>i7 : ambient T
o7 = R
o7 : QuotientRing</pre>
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<tr><td><pre>i8 : sing = singularLocus T
o8 = sing
o8 : QuotientRing</pre>
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<tr><td><pre>i9 : ideal sing
3 2 2 2 2 3 4 2 2
o9 = ideal (x + y*z, y + x*z, x y + z , z , x + y*z, x*z, x , x y, x z,
------------------------------------------------------------------------
3
x )
o9 : Ideal of S</pre>
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<tr><td><pre>i10 : ambient sing
o10 = S
o10 : PolynomialRing</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_ambient.html" title="ambient free module of a subquotient, or ambient ring">ambient</a> -- ambient free module of a subquotient, or ambient ring</span></li>
<li><span><a href="_singular__Locus.html" title="singular locus">singularLocus</a> -- singular locus</span></li>
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