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