Counterexample of prime avoidance with infinite case

Asked Sep 07 '22 at 22:14

Active Sep 08 '22 at 07:29

Viewed 81 times

The Prime Avoidance Theorem is :

Let $I \subset R$ be an ideal (with $R$ commutative ring) and $I \subseteq \bigcup_{i=1}^r P_i$, where each $P_i$ is prime. Then $I \subseteq P_i$ for some $I$.

I want to find counter example of this does not hold in the case there are infinitely many prime ideals.

I often see this example in the case of $R=\Bbb C[x,y]$, but I want to find an example in the case $R$ is Dedekind domain.

Any good ideas ?

I tried with ring of integers of number field, but in vain. Thank you for your help.

edited Sep 08 '22 at 07:29

asked Sep 07 '22 at 22:14

Pont

5,909

This seems like it would follow from unique factorization of ideals. – Kenta S Sep 08 '22 at 00:20
A quick search yields https://arxiv.org/abs/1710.05496 – Andrew Hubery Sep 08 '22 at 07:42
2

This yields a counterexample for Dedekind domains: https://math.stackexchange.com/q/51362/348926 – Lukas Heger Sep 08 '22 at 11:40

Counterexample of prime avoidance with infinite case

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