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According to Wikipedia (https://en.wikipedia.org/wiki/Euclidean_domain), the property $a\mid b\Rightarrow f(a)\leq f(b)$ is not necessary to have an Euclidean domain.

If we have a function $f$ satisfying : for all $a$ and $b$ in $R$ with $b\neq 0$, there are $q$ and $r$ in $R$ such that $a = bq + r$ and either $r = 0$ or $f(r) < f(b)$, then the function $g(x):=\min_{k\in R\setminus\{0\}}f(kx)$ satisfies both properties.

I can prove that $a\mid b\Rightarrow g(a)\leq g(b)$ but how to prove the other property for $g$?

A similar question was asked here Definition of Euclidean domain but no proof was given.

Thanks !

Friedrich
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  • A proof was given by Bill with the reference "Euclidean Rings. A. G. Agargun, C. R. Fletcher". Did you have a look already? – Dietrich Burde Aug 09 '17 at 15:22
  • Yes, I looked at the paper. It says to consider another function $h(x):=min_{u\text{ unit of }R }f(ux)$ but I don't see either how to prove that it satisfies the conditions. – Friedrich Aug 09 '17 at 15:27
  • I think there is no real proof in the reference given. – Friedrich Aug 09 '17 at 15:34
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    A proof is given in this blurb: http://www.math.uconn.edu/~kconrad/blurbs/ringtheory/euclideanrk.pdf – Xam Aug 09 '17 at 15:52
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    @Friedrich you're welcome. On the other hand, I recommend you to make a better searching, especially when the question is a very frequent one as is the case of the definition of euclidean domain. – Xam Aug 09 '17 at 16:04

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