Is $\mathbb{R}[X,Y]/(X^2+Y^2-1,X-2)$ an integral domain?

Question

Is $\mathbb{R}[X,Y]/(X^2+Y^2-1,X-2)$ an integral domain?

I know I could answer this if I were able to prove that $(X^2+Y^2-1,X-2)$ is a prime ideal (or not) or if i could prove that this is isomorphic to some other domain (tbh, i think it isn't).

Either way, I don't know where to start.

Any tips are appreciated! Thanks in advance!

Answer 1

Observe that $\mathbb{R}[X,Y]/(X-2)\simeq\mathbb{R}[Y]$ and $(X^2+Y^2-1)/(X-2)\simeq(Y^2+3)$.

By the third homomorphism theorem, we have $$\frac{\mathbb{R}[X,Y]/(X-2)}{(X^2+Y^2-1)/(X-2)}\simeq \mathbb{R}[Y]/(Y^2+3)$$

In addition to that, $(Y^2+3)$ is the kernel of the surjective homomorphism $$\varphi:R[Y]\rightarrow\mathbb{C}$$ $$Y\mapsto i\sqrt{3}$$

So, by the first homomorphism theorem, $\mathbb{R}[Y]/(Y^2+3)\simeq\mathbb{C}$. And we conclude that $\mathbb{R}[X,Y]/(X^2+Y^2-1,X-2)$ is a field, since $\mathbb{C}$ is a field, hence it's an integral domain.

Answer 2

HINT: $I$ is a prime ideal of $R$ then $R/I$ is an integral domain