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<head><title>Ring ^ List -- make a free module</title>
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<div><h1>Ring ^ List -- make a free module</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>M = R^{i,j,k,...}</tt></div>
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<li><span>Operator: <a href="_^.html" title="a binary operator, usually used for powers">^</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>R</tt>, <span>a <a href="___Ring.html">ring</a></span></span></li>
<li><span><tt>{i,j,k, ...}</tt>, <span>a <a href="___List.html">list</a></span>, of integers or lists of integers</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Module.html">module</a></span>, , a free module over <tt>R</tt> whose generators have degrees <tt>-i</tt>, <tt>-j</tt>, <tt>-k</tt>, ...</span></li>
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<div class="single"><h2>Description</h2>
<div><table class="examples"><tr><td><pre>i1 : R = QQ[a..d]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : R^{-1}
1
o2 = R
o2 : R-module, free, degrees {1}</pre>
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<tr><td><pre>i3 : R^{-1,2:-2,-3}
4
o3 = R
o3 : R-module, free, degrees {1, 2, 2, 3}</pre>
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If <tt>i</tt>, <tt>j</tt>, ... are lists of integers, then they represent multi-degrees, as in <a href="_graded_spand_spmultigraded_sppolynomial_springs.html" title="">graded and multigraded polynomial rings</a>.<table class="examples"><tr><td><pre>i4 : R = QQ[x,y,z,Degrees=>{{1,0},{1,-1},{1,-2}}]
o4 = R
o4 : PolynomialRing</pre>
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<tr><td><pre>i5 : R^{{1,2}}
1
o5 = R
o5 : R-module, free, degrees {{-1, -2}}</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_degrees.html" title="degrees of generators">degrees</a> -- degrees of generators</span></li>
<li><span><a href="_free_spmodules.html" title="">free modules</a></span></li>
<li><span><a href="_graded_spand_spmultigraded_sppolynomial_springs.html" title="">graded and multigraded polynomial rings</a></span></li>
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