I have the following definition:
Let D be an integral domain; a function $N:D \to \mathbb{N}$ is a $\textbf{Dedekind-Hasse}$ norm if $$\forall 0 \neq a,b \in D;\quad b|a \quad \lor \quad \exists s,t \in D: 0 < N(sa-tb)<N(b)$$
If we have D a PID, could we simply define:
$$N:D \to \mathbb{N} \\ \quad \quad \quad \quad x \to 1; x=0 \\ \quad \quad \quad \quad x \to 2; x \neq 0$$
If $a,b \in D - \{0\}$ and b|a, we're done, otherwise as $Da+Db$ is also an ideal, so $0=sa-tb$ for some $s,t \in D $ and $0 < N(sa-tb)<N(b)$. Is this a valid example and if so, is it of any use?