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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).

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    $\begingroup$ 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? $\endgroup$
    – Hagen Knaf
    Feb 28, 2015 at 0:34
  • $\begingroup$ ...that each element of $I$ vanishes for $X=a,Y=b$? $\endgroup$
    – Unochiii
    Feb 28, 2015 at 0:48
  • $\begingroup$ Yes, and since every element of I is a multiple of X^3-Y^2...? $\endgroup$
    – John Brevik
    Feb 28, 2015 at 1:20
  • $\begingroup$ Then in particular $a^3=b^2$, which has infinitely many solutions over the complex numbers, corresponding to infinitely many ideals of the form $(X-a,Y-b)$. $\endgroup$
    – Unochiii
    Feb 28, 2015 at 1:22

1 Answer 1

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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 $n\ge1$.

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  • $\begingroup$ Does this kind of thing have any applications? I think things of this form have something to do with algebraic geometry, but I don't know anything about that, I'm afraid. $\endgroup$
    – Unochiii
    Feb 28, 2015 at 18:50
  • $\begingroup$ @Unochiii Which "kind of thing"? $\endgroup$
    – user26857
    Feb 28, 2015 at 18:52
  • $\begingroup$ Oh, sorry! The fact that it's contained in infinitely many ideals (or prime ideals). Is this just a curiosity? $\endgroup$
    – Unochiii
    Feb 28, 2015 at 18:54
  • $\begingroup$ @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 $R/I$. $\endgroup$
    – user26857
    Feb 28, 2015 at 18:57
  • $\begingroup$ Sorry, my bad! I seemed to recall reading somewhere about leaving it a couple of days before accepting an answer. $\endgroup$
    – Unochiii
    Mar 2, 2015 at 18:43

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