Let $\mathcal{I} = (y^2 - xz, z^2 - y^3)$ and $\mathcal{J} = (x-y, xz-y^2, z-xy) \cap (z,y)$. I want to show that these ideals are equal, but am struggling to make the algebra work. I am aware that I need to write the generators of $\mathcal{I}$ as a linear combination of the generators of $\mathcal{J}$ and similarly for $\mathcal{J} \subset \mathcal{I}$.
Obviously, $y^2 - xz = -(xz - y^2)$, so this is fine. It is the other generators that I am struggling with. Thanks in advance for any of your help.
For the seond generator, you have: $z^2-y^3=z(z-xy)+y(xz-y^2)$