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There are finitely many minimal prime ideals in every $B\subseteq \operatorname{Spec} A $...

Discussion in 'Mathematics' started by Natalio, Oct 8, 2018.

  1. Natalio

    Natalio Guest

    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?

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