The kernel and coimage of a ring map can be computed using coimage and ker . The output of ker is an ideal and the output of coimage is a ring or quotient ring.
i1 : R = QQ[x,y,w]; U = QQ[s,t]/ideal(s^4+t^4);
i3 : H = map(U,R,matrix{{s^2,s*t,t^2}})
2 2
o3 = map (U, R, {s , s*t, t })
o3 : RingMap U <-- R
i4 : ker H
2 2 2
o4 = ideal (y - x*w, x + w )
o4 : Ideal of R
i5 : coimage H
R
o5 = -------------------
2 2 2
(y - x*w, x + w )
o5 : QuotientRing