One caveat with the names involved here is that since we're working in a constructive setting, we only have access to the non-Noetherian analogues to these ideas.
We can only talk about finitely generated ideals, etc. so the proper names drift into slightly more exotic territory, unique factorization domains become rather redundantly named as GCD domains, constructive principal ideal domains are Bézout domains, Dedekind domains weaken to Prüfer domains, etc.
It becomes annoyingly easy to trip up and mention something that was designed in a classical setting, and some of the constructive analogues you need lack traditional names.
-Edward
On Wed, Nov 2, 2011 at 6:56 PM, Paterson, Ross
<R.Paterson@city.ac.uk> wrote:
Tyson Whitehead writes:
> Am I correct in understanding then that there could actually be euclidean
> domains that don't have good definitions unit and associate?
The properties make sense for any integral domain; there can always be
a definition. Of course there may be some integral domains for which
the operations are not computable, just as other operations might not be.