Proving Ideal in $\mathbb{C}[x, y ,z]$ is Prime

Question

I am working on a problem, and my proof would be complete if I could show that $I = (2 + x + z, y - xz - x^2)$ is a prime ideal in $\mathbb{C}[x, y ,z]$. The methods I know (e.g., showing that $\mathbb{C}[x, y ,z] / I$ is a domain) don't seem to work well here; how can I show that $I$ is a prime ideal?

Answer 1

Hint: Prove the map $\mathbb{C}[x, y ,z] \to \mathbb{C}[x]$ given by $p(x,y,z) \mapsto p(x,x(-2 - x) +x^2,-2 - x)$ is a surjective ring homomorphism with kernel $I$.

Answer 2

We can simplify the quotient ring to $C[x,y]/J$ where $J=(xy-x^2+3y)$ by eliminating $z$. It is easy to see that $xy-x^2+3y$ is irreducible and so the ring is a domain.