On Thu, Nov 3, 2011 at 4:11 PM, Paterson, Ross
Tyson Whitehead writes:
I see an integral domain is just a commutative ring with no zero divisors (and every euclidean domain is also an 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?
Units are invertible elements, and two elements are associates if they're factors of each other. So association is an equivalence relation; in particular the associates of 1 are the units, and the only associate of 0 is itself.
Now choose a member from each association equivalence class to be the canonical associate for all the members of that class, choosing 1 as the canonical associate for the unit class. Because there are no zero divisors, that uniquely determines the canonical unit for each element.
Huh, that seems a bit arbitrary, but I can see how it would be useful. The documentation and/or laws should probably be altered to prevent the definitions 'unit = const 1' and 'associate = id' :)