Polynomial ring and maximal ideal

Question

I am really stump in this problem. Prove that $(x,y)$ and $(2,x,y)$ are prime ideals in $Z[x,y]$ but only the latter ideal is a maximal ideal.

Answer 1

Consider the ring homomorphism $\mathbf{Z}$[x,y] $\to Z$ given by $\phi(f(x,y))=f(0,0)$. Clearly $\phi$ is surjective and $ ker \phi =(x,y)$ (not hard to prove). As for the other, you can consider a homomorhism (same way) from $\mathbf{Z}$[x,y] to $\mathbf{Z}/2\mathbf{Z}$ .