Dimension localization

Question

Let $A$ be the localization of $\mathbb Z[x, y]$ in the ideal $(5, x−1, y+2)$ and $B = A/(x^2+y^2+4y−3x+6)$. Calculate the dimensions of $A$ and $B$ and study if they are regular rings.

Answer 1

First: call $X=x-1$ and $Y=y+2$. Then $\mathbb{Z}[x,y] = \mathbb{Z}[X,Y]$, so that $$A=\mathbb{Z}[X,Y]_{\langle 5,X,Y\rangle}$$ and $$x^2+y^2+4y-3x+6 = X^2+Y^2-X+3 \notin \langle 5,X,Y\rangle$$ so that $x^2+y^2+4y-3x+6$ is a unit of $A$.

This implies that $B$ is the trivial ring (which has undefined dimension).

Finally, since $\langle 5,X,Y\rangle$ is a maximal ideal of $\mathbb{Z}[X,Y]$, the dimension of $A$ is the dimension of $\mathbb{Z}[X,Y]$, i.e. $3$.