TITLE: Show that quotient rings are not isomorphic
QUESTION [9 upvotes]: I've been given a homework problem that requires me to show that the rings $\mathbb{C}[x,y]/(y - x^2)$ and $\mathbb{C}[x,y]/(xy-1)$ are not isomorphic. 

This is my attempt at a solution:
For $\mathbb{C}[x,y]/(y - x^2)$, we can parametrize in the following way: $x = t$ and $y = t^2$. Then this ring is isomorphic to $\mathbb{C}[t]$.
For $\mathbb{C}[x,y]/(xy-1)$, we can parametrize $x = t$ and $ y = 1/t$. Then this is isomorphic to $\mathbb{C}[t, 1/t]$. 
But $\mathbb{C}[t, 1/t]$ is not isomorphic to $\mathbb{C}[t]$.
Am I on the right track? If not, any helpful hints?

REPLY [12 votes]: If two rings are isomorphic, then their unit groups are isomorphic. 

In this case $\mathbb{C}[t]^{\times}\simeq \mathbb{C}^{\times}$, while $\mathbb{C}[t,t^{-1}]^{\times}\simeq\mathbb C^{\times}\times\mathbb Z$. But the groups $\mathbb C^{\times}$ and $\mathbb C^{\times}\times\mathbb Z$ are not isomorphic for an obvious reason: the first one is divisible, while the second is not.