Definition of Euclidean domain [duplicate]

abstract-algebra ring-theory definition euclidean-domain

Solution 1:

The property $\rm\,f(ab)\ge f(a)\,$ need not be assumed in order to deduce all of the basic properties of Euclidean domains. In fact, any Euclidean function can be normalized to satisfy said property by defining $\rm\:\bar f(a)\, =\, min\: f(aR^*),\ R^* = R\backslash0.\:$

Compare also the analogous Dedekind-Hasse criterion for a PID. $ $And be sure to see this paper[1], an in-depth study and comparison of a dozen different definitions/axioms for Euclidean rings.

[1] Euclidean Rings. A. G. Agargun, C. R. Fletcher
Tr. J. of Mathematics, 19, 1995, 291 - 299.

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