Can anyone give me an example of an Euclidean Domain $D$ whose Euclidean Valuation at a point $a$ , $v(a)$ is bigger than $\min \{v(ax) : x\in D\}$ ?
I know that second condition for being Euclidean Domain is superfluous . That's why I want to form a mapping from the Euclidean Domain to $\mathbb Z_{\geq0}$ , which has to be made Euclidean function by forming another function.
The following is a Euclidean valuation on $\Bbb Z$:
$$ v(a) = \begin{cases}13 &\text{if $a=5$} \\ |n| & \text{else} \end{cases}$$
To see this, let $x,y \in \Bbb Z$ with $y \neq 0$.
Then we can write $x=qy+r$ with $|r|<|y|$. If $r \neq 5$, then we are done. So suppose that $r=5$. By chaning the sign of $q$, we can assume that $y \geq 0$. So we have $5 < y$. We get that $5-y < 0$, so in particular $5-y \neq 5$, so $v(5-y)=|5-y|=y-5 < y=v(y)$, thus $x=qy+5=(q+1)y+(5-y)$ with $v(5-y)<v(y)$ which shows that $v$ is an Euclidean valuation.
For this function, we have $v(5) = 13 > 10 = v(5 \cdot 2)$, so in particular $v(5) > \min \{ v(5a) \mid a \in \Bbb Z \setminus \{0\}\}$