On 2016-01-20 12:32, Volker Braun wrote:
Of course you can define as ZZ-floor division as the above operation in QQ, but thats doesn't generalize to other rings.
True, this doesn't work in full mathematical generality. But that's not a good reason to just throw everything away.
Many useful Euclidean divisions can be defined in terms of a "floor"-like function on the fraction field.
More precisely, we could define a *floor domain* as a domain R together with a function floor: frac(R)->R such that the operation a//b, defined as floor(a/b) makes R into a Euclidean domain.
Then ZZ is a floor domain and K[x] is also a floor-domain. Also the Euclidean quadratic imaginary rings like Z[I] are floor domains.
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