Dimension of quotient ring 
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 $\mathbb{Z}$.

Let $R=\mathbb Z[X,Y]/(XY+2)$. We have $\dim R\le2$. Furthermore, since $x$ is a non-zero divisor on $R$ we have $\dim R\ge\dim R/(x)+1=2$. (Note that $R/(x)\simeq(\mathbb Z/2\mathbb Z)[Y]$.)
