Let $\mathbb{C}[x, y]$ be the ring of polynomials in 2 variables over the complex numbers and let $(xy)$ be the ideal generated by $xy$. Find all prime ideals of $\frac{\mathbb{C}[x, y]}{(xy)}$.
Attempted solution:
By nature of prime ideals, $0 \in P$ for every prime ideal $P$. Furthermore, in the quotient ring $\frac{\mathbb{C}[x, y]}{(xy)}$ we have that $xy = 0$. Therefore $xy \in P$ for every prime ideal in the quotient. By definition of prime ideals, this implies either $x$ or $y \in P$ for every prime ideal. Now, prime ideals in the quotient ring take the form $(p,f(x))$ where $p$ is prime in $\mathbb{C}$ and $f(x)$ is irreducible.
Therefore, all prime ideals in the quotient will take the form $(p,x), (p,y), (p,x,y)$ where $p = 0$ or a prime in $\mathbb{C}$.
Is this formalism correct / effectively address the question?