Generalisation of euclidean domains Recently I wondered how dependent the definition of euclidean domains is of the co-domain of the norm-function.
To be precise: Let's define a semi-euclidean domain as a domain $R$  together with a norm $\delta : R \rightarrow \alpha$ for an ordinal $\alpha$ (or even the class of all ordinals) such that for $f,g \in R\ \exists\ q,r\in R$ such that $f=qg+r$ and $\delta(r)<\delta(g)$.
With the same argument as for euclidean domains semi-euclidean domains are principal-ideal-domains. 
Are there any semi-euclidean-domains which are not euclidean?
Thanks for your ideas,
Takirion
Edit: As the image of $\delta$ is well ordered (and as such isomorphic to an ordinal) we can assume that $\delta$ is surjective. 
Here is an idea that didn't work: 
$\mathbb{R}^{\alpha}$, the "Polynomial Ring" indexed by an limit ordinal $\alpha$ with $\delta (f)=\deg(f)$, because if $\omega_1\in \alpha$ we can't divide $x^{\omega_1}$ by $x^2$.
Edit 2: Using Zorn's Lemma we can find a minimal norm in terms of the partial ordering $f\leq g \iff f(a)\leq g(a)\forall a\in R$ for two norms f and g. To show this let $(A, \leq)$ the partial ordered Set of all norms from $R$ in an ordinal $\alpha$ (note that to use Zorns Lemma here \alpha has to be an ordinal and can't be be class of all ordinals). Let $f_1\geq f_2\geq f_3\geq ...$ be a chain in $A$. Set $f(a)=min\lbrace f_n (a)|n\in \mathbb{N}\rbrace$. It's easy to see, that f is a norm on R and by definition it is a lower bound for all the $f_n$. So by Zorn's Lemma we find a minimal Element $\delta^*\in A.\square$ 
As $\delta^*$ is minimal obviously $\delta^*$ is surjective on an initialsegment of $\alpha$ (so we just assume it is surjective on $\alpha$). If $\alpha \leq \omega$ $R$ is euclidean and we are done. If not we find $x\in R$ with $\delta^*(x)=\omega$. But as $\delta^*$ is minimal this means, that for each $n\in \mathbb{N}$ we find $y\in R$ such that $y=q*x+r \implies \delta^*(r)>n$. At the moment i hope that it either will be possible to show that such a $x$ can't exist, or that this property of some elements gives hints on where to look for semi-euclidean but not euclidean rings.
I found another interesting thing: Given that there is any euclidean function on the ring, we can construct $the$ minimal norm explicitely by transfinite induction. We define $\delta^{-1}(-\infty)=0$ and $\delta^{-1}(\alpha)=\lbrace x\in R\vert\forall y\in R \exists q,r\in R:y=q*x+r, \delta(r)<\alpha \rbrace$ for every ordinal $\alpha$. If we have any euclidean function $\mu$ we see immediately that every $x\in R$ gets assigned a value $\delta(x)<\mu(x)$ which shows both, that $\delta$ is defined on all elements of $R$ and that it is minimal. That it is indeed an euclidean function follows directly from the definition. 
By all the stuff above getting assigned a limit-ordinal is a property which belongs to an element and is essentialy independent of the euclidean function.
So I guess if there is a semiceulclidean not euclidean ring those elements will be interesting objects to be studied.
 A: These 'semi-Euclidean domains' have been studied and are often called (transfinite) Euclidean Domains. Motzkin was the first to realize that the co-domain of the norm-function need only be a well ordered set (see Motzkin's paper: 'The Euclidean Algorithm') and since then many authors have looked at the similarities between the classical definition of Euclidean domains and the extended definition. (For an introduction into these types of objects see Lenstra's "Lectures on euclidean rings" here: https://www.math.leidenuniv.nl/~hwl/PUBLICATIONS/pub.html)
As for your question: "Are there any semi-euclidean domains which are not euclidean?" the answer is yes, there are transfinite Euclidean domains whose co-domain is an ordinal larger than $\omega$. Masayoshi Nagata and Jean-Jacques Hiblot independently found rings with a co-domain of $\omega^2$ (see Nagata's paper "On Euclid algorithm" or Hiblot's paper "Des anneaux euclidiens dont le plus petit algorithme n’est pas a valeurs finies").
More recently, in work done by Conidis, Nielsen and Tombs, rings were constructed to have an arbitrary indecomposable ordinal as the co-domain (see Conidis, Nielsen and Tombs' paper "Transfinitely valued Euclidean domains have arbitrary indecomposable order type").
