4
$\begingroup$

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

Any help would be greatly appreciated!

  • 3
    Do you know that the localization of a Dedekind domain is another Dedekind domain?2012-03-14

1 Answers 1

3

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

  • 1
    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