If we have $R_{i}$, $i\in I$, $I$ may be infinite and each $R_{i}$ is a Noetherian integral domain with the same quotient field $K$ then it seems $R = \bigcap_{i\in I} R_{i}$ is not necessarily Noetherian. Example is a non-Noetherian Krull domain which is an intersection of DVRs. But I can not figure it out why this is not true.
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Dear Sharma, what is it that you do not understand? You would like an example of a non-Noetherian Krull domain? – Bruno Joyal Feb 27 '13 at 23:18
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@Sharma When you speak of the intersection I think you need all your $R_i$ to lie inside of some big ring. – Feb 27 '13 at 23:20
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2@BenjaLim Well he did say "with the same quotient field $K$", which (to me) means that they are subrings of $K$... – Bruno Joyal Feb 27 '13 at 23:21
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2@Bruno Fair enough :) – Feb 27 '13 at 23:21
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If I understand your question correctly, you're looking for an example of such a collection of rings? An example I can think of would be the subrings $$R_n=k(x_{n+1},x_{n+2},\ldots)[x_1,\ldots,x_n],\quad n\in\mathbb{N}$$ of the field $K=k(x_1,x_2,\ldots)$, each of which is a polynomial ring in finitely many variables over a field, hence noetherian, and each of which has $K$ as its fraction field, but whose intersection is $$k[x_1,x_2,\ldots]$$ which is not noetherian.
Zev Chonoles
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