2
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

See Proposition 4 of the following article:

  • Pierre Samuel, About Euclidean Rings, link
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.