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Example of a Dedekind domain that has only finitely many prime ideals, and is not a field?

Any help would be greatly appreciated!

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    Do you know that the localization of a Dedekind domain is another Dedekind domain?2012-03-14

1 Answers 1

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Very simply, $\mathbb Z_{(2)} = $ ring of all rationals with odd denominator, or any DVR (local PID not a field)

Recall that Dedekind domains may be thought of as globalizations of DVRs since for local domains we have Dedekind $\iff$ PID $\iff$ DVR

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    Thank you. any chance you could explain why the ring of all rationals with odd denominator has only finitely many prime ideals?2012-03-14
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    **Hint** $\:$ Every odd prime $\:p\:$ is a unit since $\:1/p\:$ is in the ring. Conversely, $2$ is not a unit since $\rm\:1/2 = a/b\:$ $\Rightarrow$ $\: b = 2\:a\:$ is even. So only the prime $2$ survives when enlarging $\:\mathbb Z\:$ to this overring. $\quad $2012-03-14
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    It'd be nice to have an example of a Dedekind domain which has a finitely many prime ideals but is not a DVR nor a field.2012-06-05