From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: Noetherian Rings
Date: 4 Feb 2001 02:38:51 GMT
Newsgroups: sci.math
Summary: Noetherian property and localizations at primes

In article <958spo$2so$1@nnrp1.deja.com>,  <freedom641@hotmail.com> wrote:
>Let R be any commutative ring with identity element. Suppose that for
>any prime ideal P in R, the localization of R at P is Noetherian.
>Is it true that R must be itself noetherian?

Bill Blair points out to me that the answer is obviously "no";
consider direct products of copies of a field.

Earliest reference in the literature seems to be by Krull,
Math. Nachr. 21 (1960), 319--338, MR 24#A142. The term
"almost Noetherian" ("Fast-Noethersch") is used; there are
a few hits in MathSciNet.

One _can_ deduce the Noetherian property for  R  from that of all
its localizations at maximal ideals if in addition we assume each
(nonzero) element of  R  lies in only finitely many of them.

dave