|  |  5.1.116 preimage 
 
See
 ideal;
 kernel;
 map;
 ring.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 preimageis 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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