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.
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I've tried to apply the theorems/lemmas from the Krull dimension theory because I thoguht that was the right way to go but couldn't reach any relevant conclusions. It might be that I lack some knowledge, I am still looking a solution and I see that the theory of dimension in general is really vaste. I would be truly grateful for a solution! Thank you – Calin Muntean Jan 11 '16 at 16:34
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First: call $X=x1$ 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 $$x^2+y^2+4y3x+6 = X^2+Y^2X+3 \notin \langle 5,X,Y\rangle$$ so that $x^2+y^2+4y3x+6$ 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. $3$.
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Thank you very much! Honestly, I have been on a wrong path since the begining. Thanks a lot! – Calin Muntean Jan 11 '16 at 17:09

When doing these exercises, remember: the first thing to do is to shift everything to the origin. – Crostul Jan 11 '16 at 17:15