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<head><title>generators(Module) -- the generator matrix of a module</title>
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<div><h1>generators(Module) -- the generator matrix of a module</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>generators M</tt><br/><tt>gens M</tt></div>
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<li><span>Function: <a href="_generators.html" title="provide matrix or list of generators">generators</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>M</tt>, <span>a <a href="___Module.html">module</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Matrix.html">matrix</a></span>, the matrix of generators of <tt>M</tt>.</span></li>
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<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><tt>CoefficientRing => </tt><span><span>default value null</span>, unused option</span></span></li>
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<div class="single"><h2>Description</h2>
<div>Every module in Macaulay2 has, at least implicitly, a generator matrix and a matrix of relations, both of which are matrices between free modules. This function returns the generator matrix. The module is generated by (the images of) the columns of this matrix.<table class="examples"><tr><td><pre>i1 : R = GF(8,Variable=>a)
o1 = R
o1 : GaloisField</pre>
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<tr><td><pre>i2 : f = R_0 ++ R_0^2 ++ R_0^3 ++ R_0^4
o2 = | a 0 0 0 |
| 0 a2 0 0 |
| 0 0 a+1 0 |
| 0 0 0 a2+a |
4 4
o2 : Matrix R <--- R</pre>
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<tr><td><pre>i3 : generators image f
o3 = | a 0 0 0 |
| 0 a2 0 0 |
| 0 0 a+1 0 |
| 0 0 0 a2+a |
4 4
o3 : Matrix R <--- R</pre>
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<tr><td><pre>i4 : generators cokernel f
o4 = | 1 0 0 0 |
| 0 1 0 0 |
| 0 0 1 0 |
| 0 0 0 1 |
4 4
o4 : Matrix R <--- R</pre>
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<div class="single"><h2>Caveat</h2>
<div>This function returns a matrix with the given generators. This set of generators may not be minimal, or sorted in any particular order. Use <a href="_trim_lp__Module_rp.html" title="">trim(Module)</a> or <a href="_mingens_lp__Module_rp.html" title="minimal generator matrix">mingens(Module)</a> instead.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_relations.html" title="the defining relations">relations(Module)</a> -- the defining relations</span></li>
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