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<head><title>promote -- promote to another ring</title>
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<div><h1>promote -- promote to another ring</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>promote(f,R)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>f</tt>, <span>a <a href="___Ring__Element.html">ring element</a></span>, <span>an <a href="___Ideal.html">ideal</a></span>, or <span>a <a href="___Matrix.html">matrix</a></span> over some base ring of R</span></li>
<li><span><tt>R</tt>, <span>a <a href="___Ring.html">ring</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Ring__Element.html">ring element</a></span>, or <span>a <a href="___Matrix.html">matrix</a></span>, over R</span></li>
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<div class="single"><h2>Description</h2>
<div><p>Promote the given ring element or matrix <tt>f</tt> to an element or matrix of <tt>R</tt>, via the natural map to <tt>R</tt>. This is semantically equivalent to creating the natural ring map from <tt>ring f --> R</tt> and mapping f via this map.</p>
<table class="examples"><tr><td><pre>i1 : R = QQ[a..d]; f = a^2;</pre>
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<tr><td><pre>i3 : S = R/(a^2-b-1);</pre>
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<tr><td><pre>i4 : promote(2/3,S)
2
o4 = -
3
o4 : S</pre>
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<tr><td><pre>i5 : F = map(R,QQ); F(2/3)
o5 : RingMap R <--- QQ
2
o6 = -
3
o6 : R</pre>
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<tr><td><pre>i7 : promote(f,S)
o7 = b + 1
o7 : S</pre>
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<tr><td><pre>i8 : G = map(S,R); G(f)
o8 : RingMap S <--- R
o9 = b + 1
o9 : S</pre>
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<p>Promotion of real numbers to rational numbers is accomplished by using all of the bits of the internal representation.</p>
<table class="examples"><tr><td><pre>i10 : promote(101.,QQ)
o10 = 101
o10 : QQ</pre>
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<tr><td><pre>i11 : promote(.101,QQ)
3638908498915361
o11 = -----------------
36028797018963968
o11 : QQ</pre>
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<tr><td><pre>i12 : factor denominator oo
55
o12 = 2
o12 : Expression of class Product</pre>
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<tr><td><pre>i13 : ooo + 0.
o13 = .101
o13 : RR (of precision 53)</pre>
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<tr><td><pre>i14 : oo === .101
o14 = true</pre>
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<p>For promotion of ring elements, there is the following shorter notation.</p>
<table class="examples"><tr><td><pre>i15 : 13_R
o15 = 13
o15 : R</pre>
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<p>If you wish to promote a module to another ring, either promote the corresponding matrices, use the natural ring map, or use tensor product of matrices or modules.</p>
<table class="examples"><tr><td><pre>i16 : use R;</pre>
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<tr><td><pre>i17 : I = ideal(a^2,a^3,a^4)
2 3 4
o17 = ideal (a , a , a )
o17 : Ideal of R</pre>
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<tr><td><pre>i18 : promote(I,S)
2
o18 = ideal (b + 1, a*b + a, b + 2b + 1)
o18 : Ideal of S</pre>
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<tr><td><pre>i19 : m = image matrix{{a^2,a^3,a^4}}
o19 = image | a2 a3 a4 |
1
o19 : R-module, submodule of R</pre>
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<tr><td><pre>i20 : promote(gens m,S)
o20 = | b+1 ab+a b2+2b+1 |
1 3
o20 : Matrix S <--- S</pre>
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<tr><td><pre>i21 : G m
o21 = image | b+1 ab+a b2+2b+1 |
1
o21 : S-module, submodule of S</pre>
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<tr><td><pre>i22 : m ** S
o22 = cokernel {2} | a 0 |
{3} | -1 a |
{4} | 0 -1 |
3
o22 : S-module, quotient of S</pre>
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A special feature is that if <tt>f</tt> is rational, and <tt>R</tt> is not an algebra over <a href="___Q__Q.html" title="the class of all rational numbers">QQ</a>, then an element of <tt>R</tt> is provided by attempting the evident division.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_base__Rings.html" title="store the list of base rings of a ring">baseRings</a> -- store the list of base rings of a ring</span></li>
<li><span><a href="_lift.html" title="lift to another ring">lift</a> -- lift to another ring</span></li>
<li><span><a href="_liftable.html" title="whether lifting to another ring is possible">liftable</a> -- whether lifting to another ring is possible</span></li>
<li><span><a href="_substitution_spand_spmaps_spbetween_springs.html" title="">substitution and maps between rings</a></span></li>
<li><span><a href="_substitute.html" title="substituting values for variables">substitute</a> -- substituting values for variables</span></li>
<li><span><a href="___Matrix_sp_st_st_sp__Ring.html" title="tensor product">Matrix ** Ring</a> -- tensor product</span></li>
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<div class="waystouse"><h2>Ways to use <tt>promote</tt> :</h2>
<ul><li>promote(RR,type of QQ)</li>
</ul>
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