kernel(RingMap) -- kernel of a ringmap

i1 : R = QQ[a..d];

i2 : S = QQ[s,t];

i3 : F = map(S,R,{s^3, s^2*t, s*t^2, t^3})

                  3   2      2   3
o3 = map (S, R, {s , s t, s*t , t })

o3 : RingMap S <-- R

i4 : ker F

             2                    2
o4 = ideal (c  - b*d, b*c - a*d, b  - a*c)

o4 : Ideal of R

i5 : G = map(S,R,{s^5, s^3*t^2-t, s*t-s, t^5})

                  5   3 2                5
o5 = map (S, R, {s , s t  - t, s*t - s, t })

o5 : RingMap S <-- R

i6 : ker(G, SubringLimit=>1)

            2 10      2   7     2 7      2 2 4     3 5         6       3 3  
o6 = ideal(a c   + 10a b*c  - 5a c  + 25a b c  + 2a c  - 5a*b*c  - 5a*b c  +
     ------------------------------------------------------------------------
         6      5      2 3       3   2       2 3       4      2 2      3 2  
     5a*c  - a*b  + 10a b c - 15a b*c  - 5a*b c  - 5a*b  + 25a b c - 5a c  -
     ------------------------------------------------------------------------
      5    4        3      2           2     2
     c  + a  - 10a*b  + 20a b*c - 10a*b  + 5a c - 5a*b - a)

o6 : Ideal of R