Why are not the rings $F[x,y]/(y^2-x)$ and $F[x,y]/(y^2-x^2)$ isomorphic? (Where $F$ is an arbitrary field)
Roughly speaking, I realize that $F[x,y]/(y^2 -x)$ "behaves" as $F[y]$ while $F[x,y]/(y^2 -x^2)$ "behaves" as $F[y] \oplus xF[y]$, but how can I prove that with a formal argument?
Hint: the unique homomorphism from $F[y]$ to $F[x, y]/(y^2-x)$ that sends $y$ to the equivalence class $[y]$ is an isomorphism. So, as you say, $F[x, y]/(y^2-x)$ "behaves" exactly like $F[y]$ and, in particular, it is an integral domain. But, in $F[x, y]/(y^2-x^2)$, the elements $[x + y]$ and $[x - y]$ are non-zero (because the ideal $(y^2 - x^2)$ doesn't contain any polynomials of total degree less than $2$) and $[x + y]\cdot[x - y] = [(x + y)\cdot(x - y)] = [x^2 - y^2]= 0$, so $F[x, y]/(y^2-x^2)$ has non-trivial zero divisors and is not an integral domain.