Singular Manual: preimage

  | ### 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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