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5.1.116 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
See ideal; kernel; map; ring.

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