The prime ideals of the ring $\mathbb{C} \otimes \mathbb{C}$

Question

I've encountered with a problem asking to show

1. Show that every principal ideal of $\mathbb{C} \otimes \mathbb{C}$ is a prime ideal.

2. Find all prime ideals of $\mathbb{C} \otimes \mathbb{C}$.

I've just observed $\mathbb{C} \otimes \mathbb{C} = \dfrac{\mathbb{R}[x,y]}{(x^2+1,y^2+1)}$. I think its enough to prove both of the problems, but can't really see how to do it. Please help someone.

Answer 1

$\mathbb C \otimes _ {\mathbb R} \mathbb C\cong \mathbb C \otimes _{\mathbb R} \frac{\mathbb R [x]}{(x^2 +1) } \cong \frac{\mathbb C [x]}{(x^2 +1)} \cong \frac{\mathbb C [x]} {(x - i) ( x+ i) } \cong \frac{ \mathbb C [x] }{(x -i)} \times \frac{\mathbb C [x]} {(x+i)} \cong \mathbb C \times \mathbb C $

Now you see that there are exactly two prime ideals, one generated by $(1,0)$ and one by $(0,1)$, which preimages are $1 \otimes x + i \otimes 1$ and $1\otimes x + i \otimes (-1)$.

Which solves your second question. You first proposition is sadly just wrong as neither $(1)$ nor $(0)$ are prime. If you do not understand some of the isomorphisms above feel free to ask, but they are pretty natural, so you should be able to figure them out yourself.