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5.1.111 preimage
Syntax:
preimage ( map )
preimage ( ring_name, map_name, ideal_name )
preimage ( ring_name, ideal_expression, ideal_name )
Type:
- ring
ideal
Purpose:
- returns the source ring of a map (in the first case) or 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:
- As
preimage is handling ideals (not polynomials), the result of a preimage
calculation of a principal ideal is (the closure of) the preimage of the ideal,
not that of the polynomial.
Example:
| | ring r1=32003,(x,y,z,w),lp;
ring r=32003,(x,y,z),dp;
ideal i=x,y,z;
ideal i1=x,y;
ideal i0=0;
map f=r1,i;
nameof (preimage (f));
==> r1
setring r1;
ideal i1=preimage(r,f,i1);
i1;
==> i1[1]=w
==> i1[2]=y
==> i1[3]=x
// the kernel of f
preimage(r,f,i0);
==> _[1]=w
// or, use:
kernel(r,f);
==> _[1]=w
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See
ideal;
kernel;
map;
ring.
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