I want to compute the Krull Dimension of the ring $\mathbb{C}[[X,Y]]/(Y^2)$. I have tried the following and would be glad if somebody could verify or point out mistakes in the following proof:
Let $y=Y$ mod $(Y^2)$ and $f:\mathbb{C}[[X,Y]]\to \mathbb{C}[[X]][y]$ be the map sending $X\mapsto X,Y\mapsto y$. Then $\mathbb{C}[[X,Y]]/(Y^2)\cong\mathbb{C}[[X]][y]$. So the dimension of these rings are the same. But $\mathbb{C}[[X]]\subset\mathbb{C}[[X]][y]$ is an integral extension because the polynomial $T^2\in\mathbb{C}[[X]][T]$ annihilates $y$. Now we see that \begin{align*} \dim\mathbb{C}[[X]]=\dim\mathbb{C}[[X]][y]=\dim\mathbb{C}[[X,Y]]/(Y^2) \end{align*} But $\dim\mathbb{C}[[X]]=1$ because $\dim\mathbb{C}[[X]]$ is a PID (which is not a field). Thus, $\dim\mathbb{C}[[X,Y]]/(Y^2)=1$.
I think one could also use the theorem that states "$\dim A/(a)=\dim A-1$" for $(A,m)$ a local Noetherian ring and $a\in m$ a non-zero divisor, but I explicitly tried to prove it without this result.
Thank you in advance.