Ideal of $\mathbb{C}[X,Y]$ contained in infinitely many distinct proper ideals

Question

Let $R=\mathbb{C}[X,Y]$, the polynomial ring in two variables over $\mathbb{C}$, and consider the (principal) ideal $I=(X^3-Y^2)$ of $R$.

I've shown that $I$ is a prime ideal and that it is not maximal, and I'm trying now to show that it is contained in infinitely many distinct proper ideals of $R$.

There's a theorem that states that ideals of a ring $R$ containing an ideal $I$ are in bijection with ideals of $R/I$, so if I can show that the latter set is infinite then I'm done. But I'm having trouble with this (or, well, thinking about the quotient at all).

Answer 1

In fact, your claim holds over any field, not only over $\mathbb C$.

Let $K$ be a field. Then there are infinitely many proper ideals in $K[X,Y]$ containing $X^3-Y^2$.

Consider the ideals $(X^3-Y^2,X^nY^n)$ for all $n\ge1$.