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<head><title>associatedPrimes(Ideal) -- find the associated primes of an ideal</title>
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<div><a href="index.html" title="">PrimaryDecomposition</a> > <a href="_associated__Primes_lp__Ideal_rp.html" title="find the associated primes of an ideal">associatedPrimes(Ideal)</a></div>
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<div><h1>associatedPrimes(Ideal) -- find the associated primes of an 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>associatedPrimes I</tt><br/><tt>ass I</tt></div>
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<li><span>Function: <a href="../../Macaulay2Doc/html/_associated__Primes.html" title="find the associated primes of an ideal">associatedPrimes</a></span></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 (quotient of a) polynomial ring <tt>R</tt></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a list of the prime ideals in <tt>R</tt> that are associated to <tt>I</tt></span></li>
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<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="../../Macaulay2Doc/html/_associated__Primes_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html">Strategy => ...</a>, </span></li>
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<div class="single"><h2>Description</h2>
<div><tt>ass</tt> is an abbreviation for <tt>associatedPrimes</tt>.<p/>
Computes the set of associated primes for the ideal <tt>I</tt>.<table class="examples"><tr><td><pre>i1 : R = ZZ/101[a..d];</pre>
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<tr><td><pre>i2 : I = intersect(ideal(a^2,b),ideal(a,b,c^5),ideal(b^4,c^4))
4 4 2 4
o2 = ideal (b , b*c , a c )
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : associatedPrimes I
o3 = {ideal (b, a), ideal (c, b)}
o3 : List</pre>
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<table class="examples"><tr><td><pre>i4 : R = ZZ/7[x,y,z]/(x^2,x*y);</pre>
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<tr><td><pre>i5 : I=ideal(0_R);
o5 : Ideal of R</pre>
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<tr><td><pre>i6 : associatedPrimes I
o6 = {ideal(x), ideal (y, x)}
o6 : List</pre>
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<p/>
In general, the associated primes are found using Ext modules: the associated primes of codimension <tt>i</tt> of <tt>I</tt> and <tt>Ext^i(R^1/I,R)</tt> are identical, as shown in Eisenbud-Huneke-Vasconcelos, Invent. Math. 110 (1992) 207-235.<p/>
<a href="_primary__Decomposition.html" title="irredundant primary decomposition of an ideal">primaryDecomposition</a> also computes the associated primes. After doing a primaryDecomposition, calling <a href="../../Macaulay2Doc/html/_associated__Primes.html" title="find the associated primes of an ideal">associatedPrimes</a> requires no new computation, and the list of associated primes is in the same order as the list of primary components returned by <a href="_primary__Decomposition.html" title="irredundant primary decomposition of an ideal">primaryDecomposition</a>.<p/>
If the ideal is <span>a <a href="../../Macaulay2Doc/html/___Monomial__Ideal.html">monomial ideal</a></span>, then a more efficient method is used. This monomial ideal code was written by Greg Smith and Serkan Hosten. The above comments about primary decomposition hold in this case too.<table class="examples"><tr><td><pre>i7 : R = QQ[a..f];</pre>
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<tr><td><pre>i8 : I = monomialIdeal ideal"abc,bcd,af3,a2cd,bd3d,adf,f5"
2 4 3 5
o8 = monomialIdeal (a*b*c, a c*d, b*c*d, b*d , a*d*f, a*f , f )
o8 : MonomialIdeal of R</pre>
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<tr><td><pre>i9 : ass I
o9 = {monomialIdeal (a, b, f), monomialIdeal (a, d, f), monomialIdeal (b, c,
------------------------------------------------------------------------
f), monomialIdeal (b, d, f), monomialIdeal (c, d, f), monomialIdeal (a,
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c, d, f)}
o9 : List</pre>
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<tr><td><pre>i10 : primaryDecomposition I
2 5 5
o10 = {monomialIdeal (a , b, a*f, f ), monomialIdeal (a, d, f ),
-----------------------------------------------------------------------
3
monomialIdeal (b, c, f), monomialIdeal (b, d, f ), monomialIdeal (c,
-----------------------------------------------------------------------
4 3 4 5
d , d*f, f ), monomialIdeal (a, c, d , f )}
o10 : List</pre>
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The list of associated primes corresponds to the list of primary components of <tt>I</tt>: the <tt>i</tt>-th associated prime is the radical of the <tt>i</tt>-th primary component.<p><b>Original author: </b>C. Yackel, http://faculty.mercer.edu/yackel_ca/.</p>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_primary__Decomposition.html" title="irredundant primary decomposition of an ideal">primaryDecomposition(Ideal)</a> -- irredundant primary decomposition of an ideal</span></li>
<li><span><a href="../../Macaulay2Doc/html/_radical.html" title="the radical of an ideal">radical</a> -- the radical of an ideal</span></li>
<li><span><a href="../../Macaulay2Doc/html/_minimal__Primes.html" title="minimal associated primes of an ideal">minimalPrimes</a> -- minimal associated primes of an ideal</span></li>
<li><span><a href="../../Macaulay2Doc/html/_top__Components.html" title="compute top dimensional component">topComponents</a> -- compute top dimensional component</span></li>
<li><span><a href="../../Macaulay2Doc/html/_remove__Lowest__Dimension.html" title="remove components of lowest dimension">removeLowestDimension</a> -- remove components of lowest dimension</span></li>
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