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? Login To add answer/comment