Problem about the definition of Euclidean domain

Question

In the definition of domain, we first define a degree function $\vartheta: R^\times \rightarrow \mathbb{N}$ with such two constraints:

(1) $\vartheta(f)\leq \vartheta(fg)$ for all $f,g\in R^\times$.

(2) for all $f,g\in R$ with $f\in R^\times$, there exist $q,r\in R$ with $g=qf+r$ and either $r=0$ or $\vartheta(r)<\vartheta(f)$.

I wonder why we need the first constraints? I think with only the second constraint, it is enough to prove the theorem: every Euclidean ring is a PID.

Can anyone give me a example where the first constraint is used?

Wikipedia indeed says the first constraint is unnecessary in a certain sense: http://en.wikipedia.org/wiki/Euclidean_domain#Definition However, dropping it doesn't give you any extra generality. — Qiaochu Yuan, May 09 '12 at 08:23
A quibble: usually $R^\times$ would mean the units in a ring $R$, but here probably you just mean non-zero elements...? — paul garrett, Apr 30 '21 at 19:49

Bill Dubuque · Accepted Answer · 2023-03-30T04:49:12.287

From my sci.math post on 2009/7/2: The property $\rm V(a) \le V(ab)$ needn't be assumed in order to deduce all of the basic properties of Euclidean domains, since any Euclidean function can be normalized to satisfy said property by defining $\rm\:v(a) = min\: V(aD^*),\ D* = D\backslash0.\:$ This fact is so well-known it is even in the Wikipedia http://en.wikipedia.org/wiki/Euclidean_domain.

See also the analogous Dedekind-Hasse criterion for a PID. And see also this paper[1] which gives an in-depth study and comparison of a dozen different definitions/axioms for Euclidean rings.

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

Problem about the definition of Euclidean domain

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