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<div><a href="index.html" title="">PrimaryDecomposition</a> > <a href="_primary__Decomposition.html" title="irredundant primary decomposition of an ideal">primaryDecomposition</a></div>
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<div><h1>primaryDecomposition -- irredundant primary decomposition 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>primaryDecomposition I</tt></div>
</dd></dl>
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</li>
<li><div class="single">Inputs:<ul><li><span><tt>I</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span> or <span>a <a href="../../Macaulay2Doc/html/___Monomial__Ideal.html">monomial ideal</a></span> in a (quotient of a) polynomial ring <tt>R</tt>.</span></li>
</ul>
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</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, of <a href="../../Macaulay2Doc/html/___Ideal.html">ideals</a>, a minimal list of primary ideals whose intersection is <tt>I</tt></span></li>
</ul>
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</li>
<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_primary__Decomposition_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html">Strategy => ...</a>, </span></li>
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</li>
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<div class="single"><h2>Description</h2>
<div>This routine returns an irredundant primary decomposition for the ideal <tt>I</tt>. The specific algorithm used varies depending on the characteristics of the ideal, and can also be specified using the optional argument <tt>Strategy</tt>. In all cases, the radical of each entry of the output is equal to the corresponding entry of the output of <a href="../../Macaulay2Doc/html/_associated__Primes.html" title="find the associated primes of an ideal">associatedPrimes</a>.<p/>
Primary decompositions algorithms are very sensitive to their input. Some algorithms work very well on certain classes of ideals, but poorly on other classes. If this function seems to be taking too long, try another algorithm (using <a href="_primary__Decomposition_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html" title="">primaryDecomposition(..., Strategy => ...)</a>).<table class="examples"><tr><td><pre>i1 : R = QQ[a..i];</pre>
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<tr><td><pre>i2 : I = permanents(2,genericMatrix(R,a,3,3))
o2 = ideal (b*d + a*e, c*d + a*f, c*e + b*f, b*g + a*h, c*g + a*i, c*h + b*i,
------------------------------------------------------------------------
e*g + d*h, f*g + d*i, f*h + e*i)
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : C = primaryDecomposition I;</pre>
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<tr><td><pre>i4 : I == intersect C
o4 = true</pre>
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<tr><td><pre>i5 : #C
o5 = 16</pre>
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<p/>
Recall that <a href="../../Macaulay2Doc/html/___Visible__List_sp_sl_sp__Function.html" title="apply a function to elements of a list">List / Function</a> applies a function to each element of a list, returning the results as a list. This is often useful with lists of ideals, such as the list <tt>C</tt> of primary components.<table class="examples"><tr><td><pre>i6 : C/toString/print;
ideal(g,d,c,b,a,f*h+e*i)
ideal(i,h,f,e,c,b)
ideal(i,h,g,f,c,b*d+a*e)
ideal(f,e,d,c,b,a)
ideal(i,f,c,b,a,e*g+d*h)
ideal(i,f,e,d,c,b*g+a*h)
ideal(i,g,f,d,c,a)
ideal(h,e,c,b,a,f*g+d*i)
ideal(i,h,g,c,b,a)
ideal(i,h,g,e,b,c*d+a*f)
ideal(i,h,g,f,e,d)
ideal(i,h,g,d,a,c*e+b*f)
ideal(h,f,e,d,b,c*g+a*i)
ideal(g,f,e,d,a,c*h+b*i)
ideal(h,g,e,d,b,a)
ideal(f*h+e*i,c*h+b*i,f*g+d*i,e*g+d*h,c*g+a*i,b*g+a*h,c*e+b*f,c*d+a*f,b*d+a*e,a^2,i^3,h*i^2,g*i^2,e*i^2,c*i^2,b*i^2,a*i^2,g*h*i,d*f*i,c*f*i,h^3,g*h^2,b*h^2,a*h^2,d*e*h,b*e*h,g^3,a*g^2,d^2*g,a*d*g,f^3,e*f^2,d*f^2,c*f^2,b*f^2,a*f^2,d*e*f,e^3,d*e^2,b*e^2,a*e^2,d^3,a*d^2,c^3,b*c^2,a*c^2,a*b*c,b^3,a*b^2,f^2*i^2,e^2*h^2)</pre>
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<tr><td><pre>i7 : C/codim
o7 = {6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 9}
o7 : List</pre>
</td></tr>
<tr><td><pre>i8 : C/degree
o8 = {2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 86}
o8 : List</pre>
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<p/>
The corresponding list of associated prime ideals is cached in <tt>I.cache#"AssociatedPrimes"</tt>, and can be obtained by using <a href="_associated__Primes_lp__Ideal_rp.html" title="find the associated primes of an ideal">associatedPrimes(Ideal)</a>.<table class="examples"><tr><td><pre>i9 : associatedPrimes I / print;
ideal (g, d, c, b, a, f*h + e*i)
ideal (i, h, f, e, c, b)
ideal (i, h, g, f, c, b*d + a*e)
ideal (f, e, d, c, b, a)
ideal (i, f, c, b, a, e*g + d*h)
ideal (i, f, e, d, c, b*g + a*h)
ideal (i, g, f, d, c, a)
ideal (h, e, c, b, a, f*g + d*i)
ideal (i, h, g, c, b, a)
ideal (i, h, g, e, b, c*d + a*f)
ideal (i, h, g, f, e, d)
ideal (i, h, g, d, a, c*e + b*f)
ideal (h, f, e, d, b, c*g + a*i)
ideal (g, f, e, d, a, c*h + b*i)
ideal (h, g, e, d, b, a)
ideal (i, h, g, f, e, d, c, b, a)</pre>
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<div class="single"><h2>Caveat</h2>
<div>The ground ring must be a prime field.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="index.html" title="">PrimaryDecomposition</a></span></li>
<li><span><a href="_associated__Primes_lp__Ideal_rp.html" title="find the associated primes of an ideal">associatedPrimes(Ideal)</a> -- find the associated primes 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>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>primaryDecomposition</tt> :</h2>
<ul><li>primaryDecomposition(Ideal)</li>
<li>primaryDecomposition(MonomialIdeal)</li>
</ul>
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