<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- -->
<!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html -->
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" >
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head><title>Singular Book 2.1.20 -- sum, intersection, module quotient</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="___Singular_sp__Book_sp2.1.24.html">next</a> | <a href="___Singular_sp__Book_sp2.1.13.html">previous</a> | <a href="___Singular_sp__Book_sp2.1.24.html">forward</a> | <a href="___Singular_sp__Book_sp2.1.13.html">backward</a> | <a href="___M2__Singular__Book.html">up</a> | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div>
</td>
</tr>
</table>
<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_basic_spcommutative_spalgebra.html" title="">basic commutative algebra</a> > <a href="___M2__Singular__Book.html" title="Macaulay2 examples for the Singular book">M2SingularBook</a> > <a href="___Singular_sp__Book_sp2.1.20.html" title="sum, intersection, module quotient">Singular Book 2.1.20</a></div>
<hr/>
<div><h1>Singular Book 2.1.20 -- sum, intersection, module quotient</h1>
<div><table class="examples"><tr><td><pre>i1 : A = QQ[x,y,z];</pre>
</td></tr>
<tr><td><pre>i2 : M = image matrix{{x*y,x},{x*z,x}}
o2 = image | xy x |
| xz x |
2
o2 : A-module, submodule of A</pre>
</td></tr>
<tr><td><pre>i3 : N = image matrix{{y^2,x},{z^2,x}}
o3 = image | y2 x |
| z2 x |
2
o3 : A-module, submodule of A</pre>
</td></tr>
<tr><td><pre>i4 : M + N
o4 = image | xy x y2 x |
| xz x z2 x |
2
o4 : A-module, submodule of A</pre>
</td></tr>
</table>
Notice that, in Macaulay2, each module comes equipped with a list of generators, and operations such as sum do not try to simplify the list of generators.<p/>
Intersection, quotients, annihilators are found using standard notation:<table class="examples"><tr><td><pre>i5 : intersect(M,N)
o5 = image | x xy2-xz2 |
| x 0 |
2
o5 : A-module, submodule of A</pre>
</td></tr>
<tr><td><pre>i6 : M : N
o6 = ideal(x)
o6 : Ideal of A</pre>
</td></tr>
<tr><td><pre>i7 : N : M
o7 = ideal(y + z)
o7 : Ideal of A</pre>
</td></tr>
<tr><td><pre>i8 : Q = A/x^5;</pre>
</td></tr>
<tr><td><pre>i9 : M = substitute(M,Q)
o9 = image | xy x |
| xz x |
2
o9 : Q-module, submodule of Q</pre>
</td></tr>
<tr><td><pre>i10 : ann M
4
o10 = ideal(x )
o10 : Ideal of Q</pre>
</td></tr>
<tr><td><pre>i11 : M : x
o11 = image | 1 y-z x4 |
| 1 0 0 |
2
o11 : Q-module, submodule of Q</pre>
</td></tr>
</table>
</div>
</div>
</body>
</html>
