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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_ideals.html" title="">ideals</a> > <a href="_ideal_spquotients_spand_spsaturation.html" title="">ideal quotients and saturation</a></div>
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<div><h1>ideal quotients and saturation</h1>
<div><h2>colon and quotient</h2>
The <a href="_quotient.html" title="quotient or division">quotient</a> of two ideals <tt>I</tt> and <tt>J</tt> is the same as <tt>I:J</tt> and is the ideal of elements <tt>f</tt> such that <tt>f*J</tt> is contained in <tt>I</tt>.<table class="examples"><tr><td><pre>i1 : R = QQ[a..d];</pre>
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<tr><td><pre>i2 : I = ideal (a^2*b-c^2, a*b^2-d^3, c^5-d);
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : J = ideal (a^2,b^2,c^2,d^2);
o3 : Ideal of R</pre>
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<tr><td><pre>i4 : I:J
2 3 2 2 5
o4 = ideal (a*b - d , a b - c , c - d)
o4 : Ideal of R</pre>
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<tr><td><pre>i5 : P = quotient(I,J)
2 3 2 2 5
o5 = ideal (a*b - d , a b - c , c - d)
o5 : Ideal of R</pre>
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The functions <a href="__co.html" title="a binary operator, uses include repetition; ideal quotients">:</a> and <a href="_quotient.html" title="quotient or division">quotient</a> perform the same basic operation, however <tt>quotient</tt> takes two options. The first is <tt>MinimalGenerators</tt> which has default value <tt>true</tt> meaning the computation is done computing a minimal generating set. You may want to see all of the generators found, setting <tt>MinimalGenerators</tt> to <tt>false</tt> accomplishes this.<table class="examples"><tr><td><pre>i6 : Q = quotient(I,J,MinimalGenerators => false)
2 3 2 2 5 4 3 4 3
o6 = ideal (- a*b + d , - a b + c , - c + d, a c*d - a*d, a b*c*d -
------------------------------------------------------------------------
4 2 2 2 4 3 2 2 6 2 4 3 2 3
a*b*d, a b c*d - a*b , a b*c d - a*b*c , a b*c*d - a c d - a b +
------------------------------------------------------------------------
2
a*c )
o6 : Ideal of R</pre>
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<tr><td><pre>i7 : Q == P
o7 = true</pre>
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The second option is <tt>Strategy</tt>. The default is to use <tt>Iterate</tt> which computes successive ideal quotients. Currently (16 May 2001) the other possible options do not work.<h2>saturation</h2>
The saturation of an ideal <tt>I</tt> with respect to another ideal <tt>J</tt> is the ideal <tt>(I : J^*)</tt> defined to be the set of elements <tt>f</tt> in the ring such that J^N*f is contained in I for some N large enough. Use the function <a href="_saturate.html" title="saturation of ideal or submodule">saturate</a> to compute this ideal. If the ideal <tt>J</tt> is not given, the ideal <tt>J</tt> is taken to be the ideal generated by the variables of the ring <tt>R</tt> of <tt>I</tt>.<p/>
For example, one way to homogenize an ideal is to homogenize the generators and then saturate with respect to the homogenizing variable.<table class="examples"><tr><td><pre>i8 : R = ZZ/32003[a..d];</pre>
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<tr><td><pre>i9 : I = ideal(a^3-b, a^4-c)
3 4
o9 = ideal (a - b, a - c)
o9 : Ideal of R</pre>
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<tr><td><pre>i10 : Ih = homogenize(I,d)
3 2 4 3
o10 = ideal (a - b*d , a - c*d )
o10 : Ideal of R</pre>
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<tr><td><pre>i11 : saturate(Ih,d)
2 2 3 2 3 2
o11 = ideal (a*b - c*d, a c - b d, b - a*c , a - b*d )
o11 : Ideal of R</pre>
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The function <tt>saturate</tt> has three optional arguments. First a strategy for computation can be chosen. The options are , <a href="___Linear.html" title="a Strategy option value for saturate">Linear</a>, <a href="___Iterate.html" title="a Strategy option value for saturate">Iterate</a>, <a href="___Bayer.html" title="a Strategy option value for saturate">Bayer</a>, and <a href="___Eliminate.html" title="elimination order">Eliminate</a>. We leave descriptions of the options to their links, but give an example of the syntax for optional arguments.<table class="examples"><tr><td><pre>i12 : saturate(Ih,d,Strategy => Bayer)
2 2 3 2 3 2
o12 = ideal (a*b - c*d, a c - b d, b - a*c , a - b*d )
o12 : Ideal of R</pre>
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The second option is <tt>DegreeLimit => n</tt> which specifies that the computation should halt after dealing with degree n. The third option is <tt>MinimalGenerators => true</tt> which specifies that the computation should not only compute the saturation, but a minimal generating set for that ideal.</div>
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