An integral domain $R$ is a Euclidean domain iff there exists a function $N: R\setminus\{0\} \rightarrow \mathbb{Z}$ such that

  1. If $a,b\in R$, then there exists $q\in R$ such that either $a=qb$ or $N(a-qb)<N(b)$.
  2. If $a,b \in R$, then $N(a)\leq N(ab)$.

I understand that condition 2 is unnecessary, as the existence of a function satisfying condition 1 implies the existence of a function satisfying both conditions.

To understand this better, I'd like to see examples of

  1. A Euclidean function on some Euclidean domain satisfying condition 1 but not condition 2.
  2. A Euclidean domain with two Euclidean functions $f$ and $g$ satisfying conditions 1 and 2, such that the orderings $f(x)\leq f(y)$ and $g(x) \leq g(y)$ are nonisomorphic.

Can someone give nice examples of these?


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