Similar to another question I posted. Does anyone know of an example of a noetherian ring in which the integral closure of a finite extension of it's field of fractions is not noetherian.

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    $\begingroup$ you mean noetherian domain, right? $\endgroup$ – Martin Brandenburg Apr 8 '11 at 16:48
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    $\begingroup$ Do you insist that the extension is non-trivial? Nagata (Local Rings, Appendix, Ex.5) gives an example where the integral closure (in the field of fractions) is nonnoetherian. $\endgroup$ – user9072 Apr 8 '11 at 17:21
  • $\begingroup$ Although this isn't what you asked for, you may also want to check out Akizuki's counterexample (1935) of a local Noetherian domain $A$ whose integral closure inside its own field of fractions isn't a finitely generated $A$-module. You can find a modern account of this counterexample here: arxiv.org/PS_cache/alg-geom/pdf/9503/9503017v1.pdf . $\endgroup$ – user91132 Apr 8 '11 at 19:03
  • $\begingroup$ yes noetherian domain $\endgroup$ – user13953 Apr 9 '11 at 17:06

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