Let $R$ be a Euclidean Domain and $g:R\setminus\{0\}\rightarrow \mathbb{N}$ be a function such that for each $a,b\in R (b\neq 0)$ there exists $q,r\in R$ such that $a=bq+r$ and $g(r)<g(b)$ or $r=0$.

Now define $f(x)=\min_{x\in R\setminus\{0\}} g(xa)$ for all $x\in R\setminus \{0\}$.

Why is this $f$ a Euclidean function?


See Proposition 4 of the following article:

  • Pierre Samuel, About Euclidean Rings, link

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