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<head><title>map(Ring,Ring) -- make a ring map, using the names of the variables</title>
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<div><a href="_map.html" title="make a map">map</a> > <a href="_map_lp__Ring_cm__Ring_rp.html" title="make a ring map, using the names of the variables">map(Ring,Ring)</a></div>
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<div><h1>map(Ring,Ring) -- make a ring map, using the names of the variables</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>map(R,S)</tt></div>
</dd></dl>
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</li>
<li><span>Function: <a href="_map.html" title="make a map">map</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>S</tt>, <span>a <a href="___Ring.html">ring</a></span></span></li>
</ul>
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</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Ring__Map.html">ring map</a></span>, a map S --> R which maps any variable of S to a variable with the same name in R, if any, and zero otherwise</span></li>
</ul>
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</li>
<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_map_lp..._cm_sp__Degree_sp_eq_gt_sp..._rp.html">Degree => ...</a>, -- set the degree of a map</span></li>
<li><span><a href="_map_lp__Ring_cm__Ring_cm__Matrix_rp.html">DegreeLift => ...</a>, -- make a ring map</span></li>
<li><span><a href="_map_lp__Ring_cm__Ring_cm__Matrix_rp.html">DegreeMap => ...</a>, -- make a ring map</span></li>
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<div class="single"><h2>Description</h2>
<div>For example, consider the following rings.<table class="examples"><tr><td><pre>i1 : A = QQ[a..e];</pre>
</td></tr>
<tr><td><pre>i2 : B = A[x,y,Join=>false];</pre>
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<tr><td><pre>i3 : C = QQ[a..e,x,y];</pre>
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The natural inclusion and projection maps between <tt>A</tt> and <tt>B</tt> are<table class="examples"><tr><td><pre>i4 : map(B,A)
o4 = map(B,A,{a, b, c, d, e})
o4 : RingMap B <--- A</pre>
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<tr><td><pre>i5 : map(A,B)
o5 = map(A,B,{0, 0, a, b, c, d, e})
o5 : RingMap A <--- B</pre>
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</table>
The isomorphisms between B and C:<table class="examples"><tr><td><pre>i6 : F = map(B,C)
o6 = map(B,C,{a, b, c, d, e, x, y})
o6 : RingMap B <--- C</pre>
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<tr><td><pre>i7 : G = map(C,B)
o7 = map(C,B,{x, y, a, b, c, d, e})
o7 : RingMap C <--- B</pre>
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<tr><td><pre>i8 : F*G
o8 = map(B,B,{x, y, a, b, c, d, e})
o8 : RingMap B <--- B</pre>
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<tr><td><pre>i9 : oo === id_B
o9 = true</pre>
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<tr><td><pre>i10 : G*F
o10 = map(C,C,{a, b, c, d, e, x, y})
o10 : RingMap C <--- C</pre>
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<tr><td><pre>i11 : oo === id_C
o11 = true</pre>
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<p/>
The ring maps that are created are not always mathematically well-defined. For example, the map F below is the natural quotient map, but the map <tt>G</tt> is not mathematically well-defined, although we can use it in Macaulay2 to lift elements of <tt>E</tt> to <tt>D</tt>.<table class="examples"><tr><td><pre>i12 : D = QQ[x,y,z];</pre>
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<tr><td><pre>i13 : E = D/(x^2-z-1,y);</pre>
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<tr><td><pre>i14 : F = map(E,D)
o14 = map(E,D,{x, 0, z})
o14 : RingMap E <--- D</pre>
</td></tr>
<tr><td><pre>i15 : G = map(D,E)
o15 = map(D,E,{x, y, z})
o15 : RingMap D <--- E</pre>
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<tr><td><pre>i16 : x^3
o16 = x*z + x
o16 : E</pre>
</td></tr>
<tr><td><pre>i17 : G x^3
o17 = x*z + x
o17 : D</pre>
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<div class="single"><h2>Caveat</h2>
<div>The map is not always a mathematically well-defined ring map</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_substitution_spand_spmaps_spbetween_springs.html" title="">substitution and maps between rings</a></span></li>
</ul>
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