Showing an ideal is prime

Question

I have the ideal $(x-y)$ in $\mathbb{C}[x,y]$. One way is to show this ideal is prime is by showing $\mathbb{C}[x,y]/(x-y)$ is an integral domain. My gut tells me $\mathbb{C}[x,y]/(x-y) \cong \mathbb{C}[x,y]$, but I don't know how to define my isomorphism.

Answer 1

When you quotient, you're declaring the thing you're quotienting out to be 0. In your case, everything in the ideal $(y-x)$ becomes zero. Now all elements of this idea are of the form $r \cdot (y-x)$ and once $y-x$ is zero, your intuition should tell you $r \cdot (y-x)$ is zero as well. So what does $y-x=0$ mean? It means $y=x$!