Prime ideal in ring of polynomials

Question

I just read about the following problem: Let $R:=\mathbb{Z}[X,Y]$ be the polynomial ring over $\mathbb{Z}$ in two variables. Determine whether $(X^2+Y,X+Y^2)$ is a prime ideal in $R$. To do this, one can show that $\mathbb{Z}[X,Y]/(X^2+Y,X+Y^2) \cong \mathbb{Z}[X]/(X^4+X)$ and since $\mathbb{Z}[X]/(X^4+X)$ is not an integral domain, it follows that the Ideal is not prime. My question is, how do I come up with the idea of those two rings being isomorphic? Is there a trick to see such things almost immediately or is it something one has to see in order to use it? I seem to struggle with tasks such as this one, which is why a "general trick" in order to determine isomorphic rings of that sort would really help me out.

Answer 1

Generally, one can try constructing a ring homomorphism with kernel equal to the divisor and break it up using the isomorphism theorems. In this case, we can start with the homomorphism $\varphi: \mathbb{Z}[X,Y]\to \mathbb{Z}[X]$ $$\varphi(p(X, Y)) = p(X, -X^2)$$ And the kernel of this (surjective) transformation is precisely $(X^2+Y)$, and thus we get an isomorphism $\tilde{\varphi}$ $$\mathbb{Z}[X,Y]/(X^2+Y)\cong_\tilde{\varphi} \mathbb{Z}[X]$$ Now, note that we can divide the left side by $Y^2+X$ and the right side by $\tilde{\varphi}(Y^2+X) = X^4+X$ and we get $$\frac{\mathbb{Z}[X,Y]/(X^2+Y)}{(Y^2+X)}\cong \mathbb{Z}[X]/(X^4+X)$$ And finally, since $X^2 + Y$ and $Y^2 + X$ are relatively prime in $\mathbb{Z}[X,Y]$, we have $$\frac{\mathbb{Z}[X,Y]/(X^2+Y)}{(Y^2+X)}\cong \frac{\mathbb{Z}[X,Y]}{(X^2+Y, Y^2+X)}$$