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

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).

• There is a more direct way of showing what you want to show: consider the ideals $(X-a,Y-b)$, $a,b\in\mathbb{C}$. What does it mean, that $I$ is contained in such an ideal? – Hagen Knaf Feb 28 '15 at 0:34
• ...that each element of $I$ vanishes for $X=a,Y=b$? – Unochiii Feb 28 '15 at 0:48
• Yes, and since every element of I is a multiple of X^3-Y^2...? – John Brevik Feb 28 '15 at 1:20
• Then in particular $a^3=b^2$, which has infinitely many solutions over the complex numbers, corresponding to infinitely many ideals of the form $(X-a,Y-b)$. – Unochiii Feb 28 '15 at 1:22

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 $n\ge1$.
• @Unochiii I don't know how relevant it is that an ideal is contained in infinitely many proper ideals, but the prime ideals containing a given ideal $I$ are definitely useful in algebraic geometry and commutative algebra as well, they representing the Spec of the quotient ring $R/I$. – user26857 Feb 28 '15 at 18:57