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In reply to "Noetherian = local property?", posted by Mitch on Oct 30, 2005:
>What's an example of a non-noetherian commutative ring A whose localization
>at every prime ideal is noetherian?
I have an example now. It's infinite product of fields. For example if
F is any field and A = F x F x F x ... and if p is any prime ideal of the ring
A then in A_p elements look like a/b where a,b are in A, b in A\p.
If a is not in p then a/b is invertible. If a is in p then not every coordinate
of a is nonzero (since then a would be a unit) so let c be any element
whose coordinates are 1 where a is zero and 0 where a is nonzero. Then c is nonzero
and is not in p, since otherwise a + c would be in p but a+c is a unit. Therefore
a/b = ac/bc is well-defined (since c is in A\p) but ac = 0. So A_p is a field
for each prime ideal p.
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