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<head><title>Ring / Ideal -- make a quotient ring</title>
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<div><h1>Ring / Ideal -- make 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>S = R/I</tt></div>
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<li><span>Operator: <a href="__sl.html" title="a binary operator, usually used for division">/</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>
<li><span><tt>I</tt>, <span>an <a href="___Ideal.html">ideal</a></span> or element of <tt>R</tt>or <span>a <a href="___List.html">list</a></span> or <span>a <a href="___Sequence.html">sequence</a></span> of elements of <tt>R</tt></span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>S</tt>, <span>a <a href="___Quotient__Ring.html">quotient ring</a></span>, the quotient ring <tt>R/I</tt></span></li>
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<div class="single"><h2>Description</h2>
<div>If <tt>I</tt> is a ring element of <tt>R</tt> or <tt>ZZ</tt>, or a list or sequence of such elements, then <tt>I</tt> is understood to be the ideal generated by these elements. If <tt>I</tt> is a module, then it must be a submodule of a free module of rank 1.<table class="examples"><tr><td><pre>i1 : ZZ[x]/367236427846278621
ZZ[x]
o1 = ------------------
367236427846278621
o1 : QuotientRing</pre>
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<table class="examples"><tr><td><pre>i2 : A = QQ[u,v];</pre>
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<tr><td><pre>i3 : I = ideal random(A^1, A^{-2,-2,-2})
2 9 7 2 8 2 10 8 2 2 2 6 2
o3 = ideal (2u + -u*v + --v , -u + --u*v + -v , -u + u*v + -v )
7 10 7 9 3 3 5
o3 : Ideal of A</pre>
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<tr><td><pre>i4 : B = A/I;</pre>
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<tr><td><pre>i5 : use A;</pre>
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<tr><td><pre>i6 : C = A/(u^2-v^2,u*v);</pre>
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<table class="examples"><tr><td><pre>i7 : D = GF(9,Variable=>a)[x,y]/(y^2 - x*(x-1)*(x-a))
o7 = D
o7 : QuotientRing</pre>
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<tr><td><pre>i8 : ambient D
o8 = GF 9[x, y]
o8 : PolynomialRing</pre>
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The names of the variables are assigned values in the new quotient ring (by automatically running <tt>use R</tt>) when the new ring is assigned to a global variable.<p/>
Warning: quotient rings are bulky objects, because they contain a Gröbner basis for their ideals, so only quotients of <a href="___Z__Z.html" title="the class of all integers">ZZ</a> are remembered forever. Typically the ring created by <tt>R/I</tt> will be a brand new ring, and its elements will be incompatible with the elements of previously created quotient rings for the same ideal.<table class="examples"><tr><td><pre>i9 : ZZ/2 === ZZ/(4,6)
o9 = true</pre>
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<tr><td><pre>i10 : R = ZZ/101[t]
o10 = R
o10 : PolynomialRing</pre>
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<tr><td><pre>i11 : R/t === R/t
o11 = false</pre>
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