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?



Domains $D$ for which for any prime $P$, $D_P$ is a PID
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> 


