What is the dimension of the following quotient ring, $\mathbb{Z}[x,y,z]/\langle xy+2, z+4 \rangle$, where $\mathbb{Z}$ is the ring of integers?

I realized this is isomorphic to $\mathbb{Z}[x,-2/x]$. How does $-2/x$ affect the dimension since the ring is $\mathbb{Z}$.