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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_modules.html" title="">modules</a> > <a href="_free_spmodules.html" title="">free modules</a></div>
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<div><h1>free modules</h1>
<div>We use <a href="___Ring_sp^_sp__Z__Z.html" title="make a free module">Ring ^ ZZ</a> to make a new free module.<p/>
<table class="examples"><tr><td><pre>i1 : R = ZZ/101[x,y,z];</pre>
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<tr><td><pre>i2 : M = R^4
4
o2 = R
o2 : R-module, free</pre>
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Such modules are often made implicitly when constructing matrices.<table class="examples"><tr><td><pre>i3 : m = matrix{{x,y,z},{y,z,0}}
o3 = | x y z |
| y z 0 |
2 3
o3 : Matrix R <--- R</pre>
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<tr><td><pre>i4 : target m == R^2
o4 = true</pre>
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<p/>
When a ring is graded, so are its free modules. By default, the degrees of the basis elements are taken to be 0.<table class="examples"><tr><td><pre>i5 : degrees M
o5 = {{0}, {0}, {0}, {0}}
o5 : List</pre>
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We can use <a href="___Ring_sp^_sp__List.html" title="make a free module">Ring ^ List</a> to specify other degrees, or more precisely, their additive inverses.<table class="examples"><tr><td><pre>i6 : F = R^{1,4:2,3,3:4}
9
o6 = R
o6 : R-module, free, degrees {-1, -2, -2, -2, -2, -3, -4, -4, -4}</pre>
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<tr><td><pre>i7 : degrees F
o7 = {{-1}, {-2}, {-2}, {-2}, {-2}, {-3}, {-4}, {-4}, {-4}}
o7 : List</pre>
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Notice the use of <a href="__co.html" title="a binary operator, uses include repetition; ideal quotients">:</a> above to indicate repetition.<p/>
If the variables of the ring have multi-degrees represented by lists (vectors) of integers, then the degrees of a free module must also be multi-degrees.<table class="examples"><tr><td><pre>i8 : S = ZZ[a,b,c, Degrees=>{{1,2},{2,0},{3,3}}]
o8 = S
o8 : PolynomialRing</pre>
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<tr><td><pre>i9 : N = S ^ {{-1,-1},{-4,4},{0,0}}
3
o9 = S
o9 : S-module, free, degrees {{1, 1}, {4, -4}, {0, 0}}</pre>
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<tr><td><pre>i10 : degree N_0
o10 = {1, 1}
o10 : List</pre>
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<tr><td><pre>i11 : degree (a*b*N_1)
o11 = {7, -2}
o11 : List</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_graded_spmodules.html" title="">graded modules</a></span></li>
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