If the localization $R_p$ of a ring $R$ at each prime ideal $p$ in A is Noetherian, does this imply that $A$ is Noetherian?
What we call such rings which is not Noetherian but localization at each prime ideal is Noetherian ?
Can somebody provide me any counterexample of (1) and also a good reference?
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Watson
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Anoop singh
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There even exist nonNoetherian rings whose localizations at all primes are all fields. (The Boolean ring at the linked question is such a ring.) – rschwieb Jan 22 '17 at 20:00
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@user26857 Do we have some special name in mathematics literature for such rings? – Anoop singh Jan 23 '17 at 05:15
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@Anoopsingh : maybe "locally noetherian" ? This is just a guess. – Watson Jan 30 '17 at 17:04
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Consider the quotient of a polynomial ring in infinitely many variables with coefficients in a field by the ideal generated by all monomials of degree 2
Mariano Suárez-Álvarez
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