Given $I \subseteq \mathbb{R}[x,y,z]$ find the generating system of $I \cap \mathbb{R}[x,y]$

Question

I've stucked with the following problem.

Given an ideal $I=(x^2y+2xz+z^2, y^2z-2z) \subseteq \mathbb{R}[x,y,z]$. Find the generating system for an ideal $I \cap R[x,y]$ of the ring $\mathbb{R}[x,y]$.

So, I don't know, but it looks like $I \cap R[x,y]=(x^2y)$. Thut, the generating system is just $\{x^2y\}$. Am I right?

Solution is very obvious, so for me it does seems that I don't understand something.