Computing extensions of an ideal in Singular or Macaulay2

Question

Does Macaulay2 or Singular compute extensions of ideals under ring homomorphisms?

Specifically, if $\phi : R \to S$ is a ring homomorphism (say polynomial rings over $\mathbb{Q}$ which can be specified in Macaulay2 or Singular) and $I$ is an ideal in $R$ given by generators, is there a command to compute the ideal generated by $\{\phi(I)\}$ in $S$?

Answer 1

Yes, this is very easy in Macaulay2. See the code exampe below.

i1 : R = QQ[x,y]

o1 = R

o1 : PolynomialRing

i2 : I = ideal(x^2-y,y+2)

             2
o2 = ideal (x  - y, y + 2)

o2 : Ideal of R

i3 : S = QQ[s,t]

o3 = S

o3 : PolynomialRing

i5 : f = map(S,R,{s^2,t^2})

               2   2
o5 = map(S,R,{s , t })

o5 : RingMap S <--- R

i6 : f I

             4    2   2
o6 = ideal (s  - t , t  + 2)

o6 : Ideal of S