Trying to understand a quotient of a polynomial ring

Question

Is there any nice way to understand the following quotient ring: $$R=\frac{\mathbb{R}[x,y]}{(y+x^2)\cap(y-x^2-1)}$$ I think that $y+x^2$ and $y-x^2-1$ are irreducible but I'm not too familiar with polynomial rings. In particular, I am curious to see whether or not $R\cong \mathbb{R}[s]\times \mathbb{R}[t]$.

Answer 1

$R$ can't be isomorphic to a direct product of rings since it has only trivial idempotents.

Set $p_1=y+x^2$ and $p_2=y−x^2−1$. If $f$ is idempotent modulo $(p_1)∩(p_2)=(p_1p_2)$, then $p_1p_2∣f^2−f$. Since $p_i$ are irreducible (hence prime) we have $p_i∣f$ or $p_i∣f−1$. If $p_1∣f$ and $p_2∣f$ then $f=0$ in $R$. Similarly for $p_1∣f−1$ and $p_2∣f−1$. Now suppose $p_1∣f$ and $p_2∣f−1$. We get $f=p_1u$, $f−1=p_2v$, and therefore $p_1u−p_2v=1$, a contradiction. (Similarly for $p_1\mid f-1$ and $p_2\mid f$.)