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<div><h1>generators of ideals and modules</h1>
<div><div><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>L_i</tt></div>
</dd></dl>
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</li>
<li>Inputs:<ul><li><span><tt>L</tt>, <span>an <a href="___Ideal.html">ideal</a></span>, <span>a <a href="___Monomial__Ideal.html">monomial ideal</a></span>, <span>a <a href="___Module.html">module</a></span>, or <span>a <a href="___Matrix.html">matrix</a></span></span></li>
<li><span><tt>i</tt>, <span>an <a href="___Z__Z.html">integer</a></span></span></li>
</ul>
</li>
<li>Outputs:<ul><li><span><span>a <a href="___Ring__Element.html">ring element</a></span> or <span>a <a href="___Vector.html">vector</a></span> the <tt>i</tt>-th generator or column of <tt>L</tt></span></li>
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As usual in Macaulay2, the first generator has index zero.<p/>
<table class="examples"><tr><td><pre>i1 : R = QQ[a..d];</pre>
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<tr><td><pre>i2 : I = ideal(a^3, b^3-c^3, a^4, a*c);
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : numgens I
o3 = 4</pre>
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<tr><td><pre>i4 : I_0, I_2
3 4
o4 = (a , a )
o4 : Sequence</pre>
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<p/>
Notice that the generators are the ones provided. Alternatively we can minimalize the set of generators.<table class="examples"><tr><td><pre>i5 : J = trim I
3 3 3
o5 = ideal (a*c, b - c , a )
o5 : Ideal of R</pre>
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<tr><td><pre>i6 : J_0
o6 = a*c
o6 : R</pre>
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<p/>
Elements of modules are useful for producing submodules or quotients.<table class="examples"><tr><td><pre>i7 : M = cokernel matrix{{a,b},{c,d}}
o7 = cokernel | a b |
| c d |
2
o7 : R-module, quotient of R</pre>
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<tr><td><pre>i8 : M_0
o8 = | 1 |
| 0 |
o8 : cokernel | a b |
| c d |</pre>
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<tr><td><pre>i9 : M/M_0
o9 = cokernel | 1 a b |
| 0 c d |
2
o9 : R-module, quotient of R</pre>
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<tr><td><pre>i10 : N = M/(a*M + R*M_0)
o10 = cokernel | a 0 1 a b |
| 0 a 0 c d |
2
o10 : R-module, quotient of R</pre>
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<tr><td><pre>i11 : N_0 == 0_N
o11 = true</pre>
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Columns of matrices may also be used as vectors in the target module.<table class="examples"><tr><td><pre>i12 : M = matrix{{a,b,c},{c,d,a},{a-1,b-3,c-13}}
o12 = | a b c |
| c d a |
| a-1 b-3 c-13 |
3 3
o12 : Matrix R <--- R</pre>
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<tr><td><pre>i13 : M_0
o13 = | a |
| c |
| a-1 |
3
o13 : R</pre>
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<tr><td><pre>i14 : prune((image M_{1,2})/(R*M_1))
1
o14 = R
o14 : R-module, free</pre>
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<div class="single"><h2>Caveat</h2>
<div>Fewer methods exist for manipulating vectors than other types, such as modules and matrices</div>
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