Is $x$ always irreducible in the quotient ring

Question

Let $I$ be a prime ideal in $\mathbb{C}[x,y]$. Then is it true that $x + I$ is always either a unit, or irreducible?

I am trying to show that $x + I$ is irreducible for $I =(y^2-x^3-1)$, however I don't think the degree argument would work because the quotient might have an affect.

Answer 1

No. For instance, if $I=(x-f(y))$ for any polynomial $f$, then $\mathbb{C}[x,y]/I\cong \mathbb{C}[y]$ with $x$ mapping to $f(y)$, so the image of $x$ in the quotient is irreducible or a unit iff $f(y)$ is irreducible or a unit in $\mathbb{C}[y]$ (and $f(y)$ could be any element of $\mathbb{C}[y]$).