TITLE: Problem about the definition of Euclidean domain
QUESTION [9 upvotes]: 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?

REPLY [4 votes]: 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. It is true
that any Euclidean function can be normalized to satisfy said property
by defining  $\rm\:v(a) = min\: V(aD^*),\ D* = D\backslash0.\:$  This is so well-known it is
even in the Wikipedia  http://en.wikipedia.org/wiki/Euclidean_domain
Compare also the analogous Dedekind-Hasse criterion for a PID.
And be sure to see this paper[1]. It 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
Tr. J. of Mathematics, 19, 1995, 291 - 299.