@DanielFischer
Could we prove it like that?
We have that the ideal $I=\langle x-1, y+x^2-1 \rangle $ is generated by two polynomials $x-1$ and $y+x^2-1$.
Since $y+x^2-1=y+(x+1)(x-1)$ we have that $y+x^2-1=y+ \langle x-1 \rangle $.
Do we conclude from that, that $I=\langle x-1, y+x^2-1 \rangle =\langle x-1, y \rangle$ ?
Have I done something wrong? Is there a better way to prove it?