I looked at this thread but am not able to apply the answers there to my problem.
The problem is this:
Given an algebraically closed field $k$ of characteristic zero, consider the polynomial ring $k[x,y]$. Let $I = (y^2 − x^3 − x^2)$ and let $J = (xy)$ and define $A = k[x, y]/I$ and $B = k[x, y]/J$. Show that $A$ and $B$ are not isomorphic.
I don't even know where to begin, perhaps I can use some sort of parameterisation as in the other thread? If anyone can hint at how I can get started I will greatly appreciate it.
Over a field of characteristic $0$ the polynomial $y^2-x^3-x^2$ is irreducible, thus making the ideal $I$ a prime ideal and the quotient ring $A$ a domain.
On the other hand the ideal $J$ is not prime (just from the definition of prime ideal: $x$ and $y$ are not element of $J$ but their product $xy$ is) and so the quotient ring $B$ is not a domain.
Hence $A$ and $B$ cannot be isomorphic.