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Suppose $K$ is a field, and $R$ is the polynomial ring $K[x_1, \ldots, x_n]$. Suppose $S$ is a set of ideals of $R$ satisfying these properties:

  1. $S$ contains all principal ideals.
  2. If $I$, $J$, and $I \cap J$ are in $S$, then $I + J$ is in $S$.

Must $S$ contain all ideals of $R$?

I’m even more interested in the analogous question for homogenous ideals (which is stronger).

This is motivated by a larger proof I am working on. I have a property that holds for principle homogenous ideals and is closed under (2), and I want to show that it holds for all homogenous ideals.

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  • $\begingroup$ Idle remark: $R$ is a UFD so the intersection of two principal ideals is principal. So (2) implies that $S$ contains all ideals which are generated by two elements. Is it true in $R$ that if $I$ is generated by $n$ elements and $J$ is principal, then $I$ intersect $J$ is generated by (at most) $n$ elements? If it is then you're home, but it might be asking too much. $\endgroup$
    – znt
    Commented Sep 23, 2016 at 20:43
  • $\begingroup$ @znt yes, that's what I was originally thinking. I was hoping that maybe taking the LCM of the generators of I with the generator of J might give generators of $I \cap J$, but unfortunately this doesn't work (take I to be the generated by $x_1$ and $x_2$, J to be generated by $x_1 + x_2$). Anyways, I've just had success in the main proof I was working on by using a different strategy, so I no longer need an answer to this question. $\endgroup$
    – Tom Price
    Commented Sep 24, 2016 at 20:22

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