Singular Book 1.3.3 -- properties of ring maps

i1 : S = QQ[a,b,c];

i2 : R = QQ[x,y,z];

i3 : phi = map(R,S,{x,y,x^2-y^3})

                          3    2
o3 = map (R, S, {x, y, - y  + x })

o3 : RingMap R <-- S

i4 : isInjective phi

o4 = false

i5 : ker phi

            3    2
o5 = ideal(b  - a  + c)

o5 : Ideal of S

Packaged code for computing preimage is missing, but it's easy to do, as follows.

i6 : psi = map(R,S,{x,x+y,z-x^2+y^3})

                            3    2
o6 = map (R, S, {x, x + y, y  - x  + z})

o6 : RingMap R <-- S

i7 : isInjective psi

o7 = true

i8 : ker psi

o8 = ideal ()

o8 : Ideal of S