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<head><title>Hom(Module,Module) -- module of homomorphisms</title>
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<div><h1>Hom(Module,Module) -- module of homomorphisms</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>Hom(M,N)</tt></div>
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<li><span>Function: <a href="___Hom.html" title="module of homomorphisms">Hom</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>
<li><span><tt>N</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="___Module.html">module</a></span>, The module Hom_R(M,N), where M and N are both R-modules</span></li>
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<div class="single"><h2>Description</h2>
<div>If <tt>M</tt> or <tt>N</tt> is an ideal or ring, it is regarded as a module in the evident way.<p/>
<table class="examples"><tr><td><pre>i1 : R = QQ[x,y]/(y^2-x^3);</pre>
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<tr><td><pre>i2 : M = image matrix{{x,y}}
o2 = image | x y |
1
o2 : R-module, submodule of R</pre>
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<tr><td><pre>i3 : H = Hom(M,M)
o3 = subquotient (| 1 x 0 |, | -y 0 x2 0 |)
| 0 0 1 | | x 0 -y 0 |
| 0 y x | | 0 -y 0 x2 |
| 1 0 0 | | 0 x 0 -y |
4
o3 : R-module, subquotient of R</pre>
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<tr><td><pre>i4 : H1 = prune H
o4 = cokernel | x2 -y |
| -y x |
2
o4 : R-module, quotient of R</pre>
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Specific homomorphisms may be obtained using <a href="_homomorphism.html" title="get the homomorphism from element of Hom">homomorphism</a>.<table class="examples"><tr><td><pre>i5 : f1 = homomorphism H_{0}
o5 = {1} | 1 0 |
{1} | 0 1 |
o5 : Matrix</pre>
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<tr><td><pre>i6 : f2 = homomorphism H_{1}
o6 = {1} | x y |
{1} | 0 0 |
o6 : Matrix</pre>
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<tr><td><pre>i7 : f3 = homomorphism H_{2}
o7 = {1} | 0 x |
{1} | 1 0 |
o7 : Matrix</pre>
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In this example, f1 is the identity map, f2 is multiplication by x, and f3 maps x to y and y to x^2.<p/>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_homomorphism.html" title="get the homomorphism from element of Hom">homomorphism</a> -- get the homomorphism from element of Hom</span></li>
<li><span><a href="___Ext.html" title="compute an Ext module">Ext</a> -- compute an Ext module</span></li>
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