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<head><title>quotient(Ideal,Ideal) -- ideal or submodule quotient</title>
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<div><h1>quotient(Ideal,Ideal) -- ideal or submodule quotient</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>quotient(I,J)</tt><br/><tt>I:J</tt></div>
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<li><span>Function: <a href="_quotient.html" title="quotient or division">quotient</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>I</tt>, <span>an <a href="___Ideal.html">ideal</a></span>, <span>an <a href="___Ideal.html">ideal</a></span>, or <span>a <a href="___Module.html">module</a></span>, a submodule</span></li>
<li><span><tt>J</tt>, <span>an <a href="___Ideal.html">ideal</a></span>, <span>an <a href="___Ideal.html">ideal</a></span>, <span>a <a href="___Ring__Element.html">ring element</a></span>, or <span>a <a href="___Module.html">module</a></span>, a submodule</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>an <a href="___Ideal.html">ideal</a></span>, the ideal or submodule <i>I:J = {f | fJ⊂I}</i></span></li>
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<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_quotient_lp..._cm_sp__Basis__Element__Limit_sp_eq_gt_sp..._rp.html">BasisElementLimit => ...</a>, </span></li>
<li><span><a href="_quotient_lp..._cm_sp__Degree__Limit_sp_eq_gt_sp..._rp.html">DegreeLimit => ...</a>, </span></li>
<li><span><a href="_quotient_lp..._cm_sp__Minimal__Generators_sp_eq_gt_sp..._rp.html">MinimalGenerators => ...</a>, -- Decides whether quotient computes and outputs a trimmed set of generators; default is true</span></li>
<li><span><a href="_quotient_lp..._cm_sp__Pair__Limit_sp_eq_gt_sp..._rp.html">PairLimit => ...</a>, </span></li>
<li><span><a href="_quotient_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html">Strategy => ...</a>, -- Possible strategies are: Iterate, Linear, and Quotient</span></li>
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<div class="single"><h2>Description</h2>
<div>If <tt>I</tt> and <tt>J</tt> are both <a href="___Monomial__Ideal.html">monomial ideals</a>, then the result will be as well. If <tt>I</tt> and <tt>J</tt> are both submodules of the same module, then the result will be an ideal, otherwise if <tt>J</tt> is an ideal or ring element, then the result is a submodule containing <tt>I</tt>.<p/>
Gröbner bases will be computed as needed.<p/>
The colon operator <a href="__co.html" title="a binary operator, uses include repetition; ideal quotients">:</a> may be used as an abbreviation of <tt>quotient</tt> if no options need to be supplied.<p/>
If the second input <tt>J</tt> is a ring element <tt>f</tt>, then the principal ideal generated by <tt>f</tt> is used.<p/>
The computation is not stored anywhere yet, BUT, it will soon be stored under <tt>I.cache.QuotientComputation{J}</tt>, or <tt>I.QuotientComputation{J}</tt>, so that the computation can be restarted after an interrupt.<table class="examples"><tr><td><pre>i1 : R = ZZ[a,b,c];</pre>
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<tr><td><pre>i2 : F = a^3-b^2*c-11*c^2
3 2 2
o2 = a - b c - 11c
o2 : R</pre>
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<tr><td><pre>i3 : I = ideal(F,diff(a,F),diff(b,F),diff(c,F))
3 2 2 2 2
o3 = ideal (a - b c - 11c , 3a , -2b*c, - b - 22c)
o3 : Ideal of R</pre>
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<tr><td><pre>i4 : I : (ideal(a,b,c))^3
2 2
o4 = ideal (11c, 3b, 33a, b , a*b, a )
o4 : Ideal of R</pre>
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If both arguments are submodules, the annihilator of <tt>J/I</tt> (or <tt>(J+I)/I</tt>) is returned.<table class="examples"><tr><td><pre>i5 : S = QQ[x,y,z];</pre>
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<tr><td><pre>i6 : J = image vars S
o6 = image | x y z |
1
o6 : S-module, submodule of S</pre>
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<tr><td><pre>i7 : I = image symmetricPower(2,vars S)
o7 = image | x2 xy xz y2 yz z2 |
1
o7 : S-module, submodule of S</pre>
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<tr><td><pre>i8 : (I++I) : (J++J)
o8 = ideal (z, y, x)
o8 : Ideal of S</pre>
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<tr><td><pre>i9 : (I++I) : x+y+z
o9 = image | z y x 0 0 0 |
| 0 0 0 z y x |
2
o9 : S-module, submodule of S</pre>
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<tr><td><pre>i10 : quotient(I,J)
o10 = ideal (z, y, x)
o10 : Ideal of S</pre>
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<tr><td><pre>i11 : quotient(gens I, gens J)
o11 = {1} | x y z 0 0 0 |
{1} | 0 0 0 y z 0 |
{1} | 0 0 0 0 0 z |
3 6
o11 : Matrix S <--- S</pre>
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Ideal quotients and saturations are useful for manipulating components of ideals. For example, <table class="examples"><tr><td><pre>i12 : I = ideal(x^2-y^2, y^3)
2 2 3
o12 = ideal (x - y , y )
o12 : Ideal of S</pre>
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<tr><td><pre>i13 : J = ideal((x+y+z)^3, z^2)
3 2 2 3 2 2 2 2
o13 = ideal (x + 3x y + 3x*y + y + 3x z + 6x*y*z + 3y z + 3x*z + 3y*z +
-----------------------------------------------------------------------
3 2
z , z )
o13 : Ideal of S</pre>
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<tr><td><pre>i14 : L = intersect(I,J)
2 2 2 2 3 2 2 3 4 3 2 2
o14 = ideal (x z - y z , 2x y + 6x y + 6x*y + 2y - 3x z - 3x y*z + 3x*y z
-----------------------------------------------------------------------
3 4 2 2 3 4 3 2 2 3 3 2
+ 3y z, x - 6x y - 8x*y - 3y + 6x z + 6x y*z - 6x*y z - 6y z, y z )
o14 : Ideal of S</pre>
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<tr><td><pre>i15 : L : z^2
2 2 3
o15 = ideal (x - y , y )
o15 : Ideal of S</pre>
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<tr><td><pre>i16 : L : I == J
o16 = true</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_saturate.html" title="saturation of ideal or submodule">saturate</a> -- saturation of ideal or submodule</span></li>
<li><span><a href="__co.html" title="a binary operator, uses include repetition; ideal quotients">:</a> -- a binary operator, uses include repetition; ideal quotients</span></li>
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