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3.20 preimage
Syntax:
preimage ( ring_name, map_name, ideal_name )
preimage ( ring_name, ideal_expression, ideal_name )
Type:
- ideal
Purpose:
- returns the preimage of an ideal under a given map.
The second argument has to be a map from the basering to the given ring
(or an ideal defining such a map),
and the ideal has to be an ideal in the given ring.
Note:
- To compute the kernel of a map, the preimage of zero has to be determined.
Hence there is no special command for computing the kernel of a map in
SINGULAR.
Note:
- at the moment it could be computed
only for maps A -> B, where A is commutative
Example:
| LIB "ncalg.lib";
ring R = 0,a,dp;
def Usl2 = sl2();
setring Usl2;
poly C = 4*e*f+h^2-2*h; // the central element of U(sl2)
ideal I = e^3,f^3,h^3-4*h;
ideal J = twostd(I); // two-sided GB
ideal K = std(I); // left GB
map Phi = R,C;
setring R;
ideal PreJ = preimage(Usl2,Phi,J); // central character of J
PreJ;
==> PreJ[1]=a2-8a
factorize(PreJ[1],1); // there are two characters for J
==> _[1]=a
==> _[2]=a-8
ideal PreK = preimage(Usl2,Phi,K); // central character of K
PreK;
==> PreK[1]=a3-32a2+192a
factorize(PreK[1],1); // there are three characters for K
==> _[1]=a
==> _[2]=a-8
==> _[3]=a-24
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See
ideal;
map;
ring.
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