TITLE: Showing two polynomial rings over $\mathbb{C}$ aren't isomorphic
QUESTION [5 upvotes]: Im trying to show that the ring of polynomials in one variable over the complex numbers is not isomorphic to the ring over $\mathbb C$ with two variables $x$ and $y$ modulo $\langle x^2-y^3\rangle$. I've shown previously that if the relationship $p^2=q^3$ holds for some $p$ and $q$ in one variable, there exists $r$ such that $p=r^3$ and $q=r^2$. I'm assuming this helps in some way but I'm not precisely sure how.

REPLY [3 votes]: A simple one:
Let $\def\CC{\mathbb C}A=\CC[X,Y]/(X^2-Y^3)$ and let consider the ideal $J=(X,Y)$ of $A$, which is maximal. One can easily see that if $d:A\to A$ is a derivation, then $d(J)\subseteq J$.
On the other hand, let $B=\CC[T]$ and consider the derivation $\delta:f\in B\mapsto f'\in B$ . Then for no maximal ideal $I\subseteq B$ we have $\delta(I)\subseteq I$. indeed,  given such an ideal there is an $\alpha\in C$ such that $I=(T-\alpha)$, and then $I\not\ni1=\delta(T-\alpha)\in \delta(I)$.
It follows that $A\not\cong B$.