proof that a Noetherian domain is Dedekind if it is locally a PID
We show that for a Noetherian (http://planetmath.org/Noetherian) domain R with field of fractions
k, the following are equivalent
.
-
1.
R is Dedekind. That is, it is integrally closed
and every nonzero prime ideal
is maximal.
- 2.
For a given maximal ideal 𝔪 and ideal 𝔞 of R, we shall write ¯𝔞 for the ideal generated by 𝔞 in R𝔪, which consists of the elements of the form s-1a for a∈𝔞 and s∈R∖𝔪. It is then easily seen that 𝔭↦¯𝔭 gives a bijection between the prime ideals of R contained in 𝔪 and the prime ideals of R𝔪, with inverse 𝔭↦R∩𝔭. In particular ¯𝔪 is the unique maximal ideal of R𝔪, which is therefore a local ring
.
Now suppose that R is Dedekind, then the localization R𝔪 will be a Dedekind domain
(localizations of Dedekind domains are Dedekind) with a unique maximal ideal, so it is a principal ideal domain (Dedekind domains with finitely many primes are PIDs).
Only the converse remains to be shown, so suppose that R is a Noetherian domain such that R𝔪 is a principal ideal domain for every maximal ideal 𝔪. In particular, R𝔪 is integrally closed and every nonzero prime ideal is maximal, so it contains a unique nonzero prime ideal ¯𝔪.
We start by showing that every nonzero prime ideal 𝔭 of R is maximal. Choose a maximal ideal 𝔪 containing 𝔭. Then, ¯𝔭 is a nonzero prime ideal, so ¯𝔭=¯𝔪 and therefore 𝔭=𝔪 is maximal.
We finally show that R is integrally closed. So, choose any x integral over R and lying in its field of fractions. Let 𝔞 be the ideal
𝔞={a∈R:ax∈R}. |
We use proof by contradiction to show that 𝔞 is the whole of R. So, supposing that this is not the case, there exists a maximal ideal 𝔪 containing 𝔞. Then x will be integral over the integrally closed ring R𝔪 and therefore x∈R𝔪. So, x=s-1y for some s∈R∖𝔪 and y∈R. Then, sx=y∈R so s∈𝔞⊆𝔪, which is the required contradiction
. Therefore, 𝔞=R and, in particular, 1∈𝔞 and x=1x∈R, showing that R is integrally closed.
Title | proof that a Noetherian domain is Dedekind if it is locally a PID |
---|---|
Canonical name | ProofThatANoetherianDomainIsDedekindIfItIsLocallyAPID |
Date of creation | 2013-03-22 18:35:27 |
Last modified on | 2013-03-22 18:35:27 |
Owner | gel (22282) |
Last modified by | gel (22282) |
Numerical id | 4 |
Author | gel (22282) |
Entry type | Proof |
Classification | msc 13F05 |
Classification | msc 13A15 |