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Does there exist a prime ideal containing all but one minimal prime ideal in an unital commutative Noetherian ring $A$, for any given minimal prime ideal $\mathfrak{p}$?

I encountered this statement while trying to prove the finiteness of the set of minimal prime ideals of an unital commutative Noetherian ring.

Any help is appreciated.

Yours gratefully,

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No. Let $k$ be a field and consider $R=k\times k \times k$. The prime ideals of $R$ are $k\times k \times 0$, $k\times 0\times k$, and $0\times k\times k$, so no prime ideal contains all but one minimal prime.

Verifying the claim that a noetherian ring has finitely many minimal primes can be done in lots of different ways depending on your preferences. I'd suggest a quick search of MSE or the web to find proofs that you like - for instance, as a geometer I like something along the lines of 00FR, but those who are more commutative algebra minded may prefer a different approach.

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