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kernel and image of a ring map

The kernel and image of a ring map can be computed using image and kernel . The output of ker is an ideal and the output of image is a ring or quotient ring.
i1 : R = QQ[x,y,w]; U = QQ[s,t,u]/ideal(s^2);
i3 : H = map(U,R,matrix{{s^2,t^3,u^4}})

                     3   4
o3 = map (U, R, {0, t , u })

o3 : RingMap U <--- R
i4 : ker H

o4 = ideal x

o4 : Ideal of R