On November 2, 2011 18:56:41 Paterson, Ross 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.
That's okay. I wasn't so interested in whether it was computable or not. I was just trying to get a feel for the nature of the structure. I see an integral domain is just a commutative ring with no zero divisors (and every euclidean domain is also an integral domain) http://en.wikipedia.org/wiki/Integral_domain If I'm understanding you then this is sufficient structure to tell us that an associate and unit decomposition exists, even if we can't compute it. I spent sometime last night trying to figure out what about this structure guarantees such a decomposition. I didn't have much luck though. Any hints? Thanks! -Tyson