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<div><h1>maps between modules</h1>
<div>Maps between free modules are usually specified as matrices, as described in the section on <a href="_matrices.html" title="">matrices</a>. In this section we cover a few other techniques.<p/>
Let's set up a ring, a matrix, and a free module.<table class="examples"><tr><td><pre>i1 : R = ZZ/101[x,y,z];</pre>
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<tr><td><pre>i2 : f = vars R
o2 = | x y z |
1 3
o2 : Matrix R <--- R</pre>
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<tr><td><pre>i3 : M = R^4
4
o3 = R
o3 : R-module, free</pre>
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We can use <a href="___Module_sp^_sp__List.html" title="projection onto summand">Module ^ List</a> and <a href="___Module_sp_us_sp__List.html" title="map from free module to some generators">Module _ List</a> to produce projection maps to quotient modules and injection maps from submodules corresponding to specified basis vectors.<table class="examples"><tr><td><pre>i4 : M^{0,1}
o4 = | 1 0 0 0 |
| 0 1 0 0 |
2 4
o4 : Matrix R <--- R</pre>
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<tr><td><pre>i5 : M_{2,3}
o5 = | 0 0 |
| 0 0 |
| 1 0 |
| 0 1 |
4 2
o5 : Matrix R <--- R</pre>
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<p/>
Natural maps between modules can be obtained with <a href="_induced__Map.html" title="compute an induced map">inducedMap</a>; the first argument is the desired target, and the second is the source.<table class="examples"><tr><td><pre>i6 : inducedMap(source f, ker f)
o6 = {1} | -y 0 -z |
{1} | x -z 0 |
{1} | 0 y x |
o6 : Matrix</pre>
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<tr><td><pre>i7 : inducedMap(coker f, target f)
o7 = | 1 |
o7 : Matrix</pre>
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