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Let $A$ be a Noetherian ring, and let $n$ be a fixed natural number. Show that there are only finitely many primes $P$ of $A$ such that the cardinality of $A/P \leq n$.

Prime ideals in noetherian rings have a property: if we have one prime ideal strictly in between two prime ideals, then we have infinitely many primes between them. Noetherian rings and prime ideals

Is this useful here? I don't know how to start, kindly help.

Mr.Multitalented
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  • I don't see any reason to think that fact would be helpful. I would do this by mapping $A$ to the product of all those quotients and then showing the image of $A$ in the product is generated by idempotents and thus zero-dimensional. Maybe there is a more direct argument though. – Eric Wofsey Nov 30 '22 at 03:59
  • can you elaborate more or maybe write it as an answer? – Mr.Multitalented Nov 30 '22 at 05:52
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    In fact, this is true not just for prime ideals of $A$, but all ideals of $A$. See the beautiful answer here: https://math.stackexchange.com/a/3796268/73817 – Alex Wertheim Nov 30 '22 at 06:46

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