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 imageis 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