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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_modules.html" title="">modules</a> > <a href="_submodules_spand_spquotients.html" title="">submodules and quotients</a></div>
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<div><h1>submodules and quotients</h1>
<div><h2>submodules</h2>
We can create submodules by using standard mathematical notation, keeping in mind that the generators of a module <tt>M</tt> are denoted by <tt>M_0, M_1</tt>, etc.<table class="examples"><tr><td><pre>i1 : R = QQ[x,y,z];</pre>
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<tr><td><pre>i2 : M = R^3
3
o2 = R
o2 : R-module, free</pre>
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<tr><td><pre>i3 : I = ideal(x^2,y^2-x*z)
2 2
o3 = ideal (x , y - x*z)
o3 : Ideal of R</pre>
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Here are some examples of submodules of <tt>M</tt>.<table class="examples"><tr><td><pre>i4 : I*M
o4 = image | x2 0 0 y2-xz 0 0 |
| 0 x2 0 0 y2-xz 0 |
| 0 0 x2 0 0 y2-xz |
3
o4 : R-module, submodule of R</pre>
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<tr><td><pre>i5 : R*M_0
o5 = image | 1 |
| 0 |
| 0 |
3
o5 : R-module, submodule of R</pre>
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<tr><td><pre>i6 : I*M_1
o6 = image | 0 0 |
| x2 y2-xz |
| 0 0 |
3
o6 : R-module, submodule of R</pre>
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<tr><td><pre>i7 : J = I*M_1 + R*y^5*M_1 + R*M_2
o7 = image | 0 0 0 0 |
| x2 y2-xz y5 0 |
| 0 0 0 1 |
3
o7 : R-module, submodule of R</pre>
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To determine if one submodule is contained in the other, use <a href="_is__Subset_lp__Module_cm__Module_rp.html" title="whether one object is a subset of another">isSubset(Module,Module)</a>.<table class="examples"><tr><td><pre>i8 : isSubset(I*M,M)
o8 = true</pre>
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<tr><td><pre>i9 : isSubset((x^3-x)*M,x*M)
o9 = true</pre>
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Another way to construct submodules is to take the kernel or image of a matrix.<table class="examples"><tr><td><pre>i10 : F = matrix{{x,y,z}}
o10 = | x y z |
1 3
o10 : Matrix R <--- R</pre>
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<tr><td><pre>i11 : image F
o11 = image | x y z |
1
o11 : R-module, submodule of R</pre>
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<tr><td><pre>i12 : kernel F
o12 = image {1} | -y 0 -z |
{1} | x -z 0 |
{1} | 0 y x |
3
o12 : R-module, submodule of R</pre>
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The module <tt>M</tt> does not need to be a free module. We will see examples below.<h2>quotients</h2>
If N is a submodule of M, construct the quotient using <a href="___Module_sp_sl_sp__Module.html" title="quotient module">Module / Module</a>.<table class="examples"><tr><td><pre>i13 : F = R^3
3
o13 = R
o13 : R-module, free</pre>
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<tr><td><pre>i14 : F/(x*F+y*F+R*F_2)
o14 = cokernel | x 0 0 y 0 0 0 |
| 0 x 0 0 y 0 0 |
| 0 0 x 0 0 y 1 |
3
o14 : R-module, quotient of R</pre>
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When constructing M/N, it is not necessary that M be a free module, or a quotient of a free module. In this case, we obtain a subquotient module, which we describe below.</div>
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