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<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_sp2.1.10.html" title="Submodules of A^n">Singular Book 2.1.10</a></div>
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<div><h1>Singular Book 2.1.10 -- Submodules of A^n</h1>
<div>A common method of creating a submodule of A^n in Macaulay2 is to take the image of a matrix. This will be a submodule generated by the columns of the matrix.<table class="examples"><tr><td><pre>i1 : A = QQ[x,y,z];</pre>
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<tr><td><pre>i2 : f = matrix{{x*y-1,y^4},{z^2+3,x^3},{x*y*z,z^2}}
o2 = | xy-1 y4 |
| z2+3 x3 |
| xyz z2 |
3 2
o2 : Matrix A <--- A</pre>
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<tr><td><pre>i3 : M = image f
o3 = image | xy-1 y4 |
| z2+3 x3 |
| xyz z2 |
3
o3 : A-module, submodule of A</pre>
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<tr><td><pre>i4 : numgens M
o4 = 2</pre>
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<tr><td><pre>i5 : ambient M
3
o5 = A
o5 : A-module, free</pre>
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A submodule can easily be moved to quotient rings.<table class="examples"><tr><td><pre>i6 : Q = A/(x^2+y^2+z^2);</pre>
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<tr><td><pre>i7 : substitute(M,Q)
o7 = image | xy-1 y4 |
| z2+3 -xy2-xz2 |
| xyz z2 |
3
o7 : Q-module, submodule of Q</pre>
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