Background
Definition 1: Let $R$ be an integral domain. A mapping $\theta:R\to\Bbb{N}$ is called Dedekind-Hasse norm on $R$ if
$(i)$ $\theta(rs)=\theta(r)\theta(s)$ for all $r,s\in R$ ($\theta$ is a norm).
$(ii)$ $\theta >0$ for all $r\in R\setminus \{0\}$ ($\theta$ is positive).
$(iii)$ for $r,s\in R$, either $s$ divides $r$ or there are $x,y\in R$ such that $0<\theta(rx-sy)<\theta(s)$.
$\quad$ Note that with $s=1$, this is a Euclidean norm.
Definition 2 An integral domain $R$ is a Euclidean domain if there is a function $\delta$ from the nonzero elements of $R$ to the nonnegative integers with these properties;
(i) If $a$ and $b$ are nonzero elements of $R$, then $\delta(a)\leq \delta(ab)$.
(ii) If $a,b\in R$ and $b\neq 0_R$, then there exist $q,r\in R$ such that $a=bq+r$ and either $r=0_R$ or $\delta(r) < \delta(b)$.
Question
I want to ask the motivation behind Definition 1, the dedekind hasse norm. I understand that the Euclidean norm for Definition 2 has to do with conditions for allowing for euclidean algorithm in integral domains while for the case of Dedekind Hasse Norm, it more have to do with integral domain allowing for GCD? (I am guessing h ere since $rx-sy$ reminds me of gcd). Also, can someone exhibit an example of integral domain that satisfy and a non-example of an integral domain that violates conditions of the Dedekind Hasse Norm. I know the Dedekind hasse norm is applied showing that $\Bbb{Z}[\frac{1+\sqrt{-19}}{2}]$ is not a euclidean domain, or that if an integral domain is a PID iff it satisfies the conditions for the Dedekind Hasse Norm. But I am looking for examples of explicit integral domain like in undergraduate algebra exercises showing whether a particular integral domain is not a Dedekind Hasse norm or an integral domain with a particular $\theta$ function satisfies the Dedekind Hasse norm.
Thank you in advance.
