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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_substitution_spand_spmaps_spbetween_springs.html" title="">substitution and maps between rings</a> > <a href="_basic_spconstruction_cm_spsource_spand_sptarget_spof_spa_spring_spmap.html" title="">basic construction, source and target of a ring map</a></div>
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<div><h1>basic construction, source and target of a ring map</h1>
<div><h2>constructing a ring map</h2>
Use the function <a href="_map.html" title="make a map">map</a> construct a map between two rings. The input, in order, is the target, the source, and the images of the variables of the source ring. The images can be given as a matrix or a list.<table class="examples"><tr><td><pre>i1 : S = QQ[x,y,z]/ideal(x^3+y^3+z^3);</pre>
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<tr><td><pre>i2 : T = QQ[u,v,w]/ideal(u^3+v^3+w^3);</pre>
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<tr><td><pre>i3 : G = map(T,S,matrix{{u,v,w^2}})
2
o3 = map(T,S,{u, v, w })
o3 : RingMap T <--- S</pre>
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<tr><td><pre>i4 : G(x^3+y^3+z)
6 2
o4 = - w + w
o4 : T</pre>
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If the third argument is not given there are two possibilities. If a variable in the source ring also appears in the target ring then that variable is mapped to itself and if a variable does not appear in the target ring then it is mapped to zero.<table class="examples"><tr><td><pre>i5 : R = QQ[x,y,w];</pre>
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<tr><td><pre>i6 : F = map(S,R)
o6 = map(S,R,{x, y, 0})
o6 : RingMap S <--- R</pre>
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<tr><td><pre>i7 : F(x^3)
3 3
o7 = - y - z
o7 : S</pre>
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<h2>source and target</h2>
Once a ring map is defined the functions <a href="_source.html" title="source of a map">source</a> and <a href="_target.html" title="target of a map">target</a> can be used to find out what the source and target of a map are. These functions are particularly useful when working with matrices (see the next example). <table class="examples"><tr><td><pre>i8 : U = QQ[s,t,u, Degrees => {{1,2},{1,1},{1,3}}];</pre>
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<tr><td><pre>i9 : H = map(U,R,matrix{{s^2,t^3,u^4}})
2 3 4
o9 = map(U,R,{s , t , u })
o9 : RingMap U <--- R</pre>
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<tr><td><pre>i10 : use R; H(x^2+y^2+w^2)
8 6 4
o11 = u + t + s
o11 : U</pre>
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<tr><td><pre>i12 : source H
o12 = R
o12 : PolynomialRing</pre>
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<tr><td><pre>i13 : target H
o13 = U
o13 : PolynomialRing</pre>
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<h2>obtaining the matrix defining a map</h2>
Use <tt>F.matrix</tt> to obtain the matrix defining the map F.<table class="examples"><tr><td><pre>i14 : H.matrix
o14 = | s2 t3 u4 |
1 3
o14 : Matrix U <--- U</pre>
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<tr><td><pre>i15 : source H.matrix
3
o15 = U
o15 : U-module, free</pre>
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<tr><td><pre>i16 : target H.matrix
1
o16 = U
o16 : U-module, free</pre>
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For more on matrices from maps see <a href="_inputting_spa_spmatrix.html" title="">inputting a matrix</a>.</div>
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