Is there any name or alternative characterization for the class of integral domains $D$ such that for any prime ideal $P$, $D_P$ is a principal ideal domain?
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
2
It's called an "almost Dedekind domain" in the literature on non-Noetherian commutative algebra. Every almost Dedekind domain is a Prüfer domain, or equivalently, locally a valuation domain. However, there exist Prüfer domains that are not almost Dedekind, e.g. any valuation domain that's not a PID. Both classes of domains come up a lot in the literature.
http://www.ams.org/journals/proc/1964-015-05/S0002-9939-1964-0166212-8/S0002-9939-1964-0166212-8.pdf
-
$\begingroup$ It's not hard to see that the rings mentioned in the question are Prüfer domains, since the image of any ideal (and hence image of every f.g. ideal) is principal, so every f.g. ideal is invertible, hence Prüfer. However, the rings defined in paper by Gilmer are those for which the localisation at every maximal ideal is Dedekind, not principal, and I can't immediately see why these are equivalent, so the class in this answer is potentially more general. Can anyone explain this or offer a counterexample? $\endgroup$– user1437Commented Dec 30, 2013 at 15:23
-
$\begingroup$ A local ring is Dedekind if and only if it is a DVR if and only if it is a PID. $\endgroup$ Commented Dec 30, 2013 at 21:57
$\begingroup$
$\endgroup$
2
If you add the hypothesis that $D$ is noetherian, then this is one of the characterizations of Dedekind rings.
-
$\begingroup$ yes this is well known. This is why I am considering the class mentioned above. $\endgroup$– Robert MCommented Dec 29, 2013 at 10:34
-
1$\begingroup$ I was suspecting that. Then I think all you can say is that your class of rings is a special class of Prüfer rings. $\endgroup$– abxCommented Dec 29, 2013 at 10:39