Definition talk:GCD Domain

Euclidean Domain

Is there a difference between a GCD domain and a Euclidean domain? From Euclidean Domain is GCD Domain it is clear that a Euclidean domain is a GCD domain, but does it always hold the other way about? Looking at it, I feel intuitively that it may not. WeedyPedia doesn't comment.

Yet another refinement in that great classification of Abstract Algebraic structures. This is fun. Rock on. --prime mover 16:21, 8 May 2011 (CDT)

A Euclidean domain is necessarily a PID, while a GCD domain is a non-Noetherian UFD. Something like this:

$\text{Euclidean Domains} \subseteq \text{PIDs} \subseteq \text{UFDs} \subseteq \text{GCD Domains}$

e.g. $k[X,Y]$ is a GCD-domain but not Euclidean. Theres also EL-domains (Euclid's Lemma), pretty much I think the rule is: if you append "-domain" to a mnemonic for some elementary number theoretic result, then write a paper with 40-odd classifications for such things, then it becomes a valid algebraic object. --Linus44 10:18, 11 May 2011 (CDT)

The biggest kudos is for demonstrating a counterexample to the suggestion that an inclusion might be bidirectional. It's fun, a bit like trainspotting or stamp-collecting. Until someone proves some category-theoretical result which completely drains the marsh. --prime mover 13:56, 11 May 2011 (CDT)