Let $R$ be a commutative ring with $1$ with the property that for any infinite family $\{P_i\}_{i\in I}$ of distinct prime ideals of $R$ we have $\cap_{i\not= j} P_i\subseteq P_j$ for all but fnitely many $j\in I$. Is there any characterization for such a ring?
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$\begingroup$ I suggest revising the title to include some keywords such as "infinite prime avoidance." $\endgroup$– Joshua MundingerJul 16, 2020 at 15:55
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$\begingroup$ @Steven Landsburg: Clearly the statement is true if $P_i's$ are not distinct and since I have stated ** for any infinite family=for every infinite family** it means that we are working on an arbitrary family of distinct prime ideals. $\endgroup$– A. C. BillerJul 17, 2020 at 0:09
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$\begingroup$ I have the following obvious/extreme examples and counter-examples in mind and I am wondering if you have discovered richer and subtler illustrations to work with. Examples: (1) Finite rings, valuation rings (more generally uniserial rings) (2) UFDs, Dedekind domains. Counter-examples: rings which surject onto a direct product of infinitely many rings. I am also wondering whether anything can be said about meet-irreducible rings (en.wikipedia.org/wiki/Irreducible_ring). $\endgroup$– Luc GuyotJul 22, 2020 at 20:13
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