I'm studying basic Ring Theory. And in my textbook, the author states the definition of Euclidean domain:
An integral domain $R$ is called to be a Euclidean domain precisely when there is a function $f: R\setminus\{0\}\rightarrow\Bbb N_0$, called degree function of $R$, such that:
(i) If $a,b \in R\setminus\{0\}$ and there exists $c\in R$ such that $ac=b$ then $f(a)\le f(b)$.
(ii) $a,b\in R$ with $b\neq 0$, then there exist $q,r\in R$ such that
$a=bq + r$ with $r=0$ or $r\neq 0$ and $f(r)\lt f(b)$.
I know the fact: all units in $R$ have smallest degree, and this question pops into my head:
I want to prove that all units have degree $0$.
Unfortunately I have no idea for it. Can anyone has an answer for my question or give me a readable explanation about it? I really appreciate !