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It is a well known fact that for a noetherian ring $A$ there are finitely many minimal primes. Now I'm wondering if this is true for every subset in the primes of $A$. My question, specifically, would be if for every non empty $B\subseteq \operatorname{Spec} A $ the set $X=\{P\in B\mid P\text{ is minimal in } B\} $ is finite.

I came up with this question when trying to prove something about $\operatorname{Supp}M$, where $M $ is an $A$-module. Perhaps if the answer to the first question is no, it turn out to be yes when $B=\operatorname{Supp}M $.

I couldn't find nothing about it on internet. Any ideas?

user26857
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Natalio
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    This is surely true for closed subsets $B$, since they correspond to quotients of $A$. If $B$ is not closed, you can cook up strange things, for example take $B$ to be the set of all closed points in an infinite $\mathrm{Spec}(A)$. – darko Oct 08 '18 at 14:16
  • @darko So when $B=\operatorname{supp}M $ the result is true since it is closed, right? How do you prove that? – Natalio Oct 08 '18 at 14:32

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Let $A$ be a noetherian ring with infinitely many maximal ideals (e.g., $A=K[x]$, where $K$ is an infinite field).

Now let $B$ be the set of maximal ideals of $A$.

Clearly all elements of $B$ are minimal in $B$.

As to your question "Ok, but what happen when it is closed?", suppose $B$ is a closed subset of ${\text{Spec}}\; A$.

Then $B=V(I)=\{P\in {\text{Spec}}\; A\mid P\supseteq I\}$ for some ideal $I$ of $A$.

Note that the minimal elements of $B$ correspond to the minimal prime ideals of $A/I$, hence there are at most fininitely many, since $A$ noetherian implies $A/I$ is noetherian.

quasi
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  • Thank you very much! As some one pointed out in the comments, the counterexample should be a non closed subset, as it is yours. Is it true if $B $ is closed? For instance when $B=\operatorname{Supp}M $ which is closed. – Natalio Oct 08 '18 at 14:36
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    If $M$ is not finitely generated, the support need not be closed. See this answer: https://math.stackexchange.com/questions/111214/example-of-a-module-whose-support-is-not-closed – quasi Oct 08 '18 at 14:39
  • Ok, but what happen when it is closed? – Natalio Oct 08 '18 at 14:44