Applications of the Dedekind-Hasse criterion

Question

It is a fact that an integral domain $R$ is a principal ideal domain if and only if there is a Dedekind-Hasse function $|R|\setminus\{0\}\xrightarrow{\ \ \delta\ \ }\mathbb{N}$ on $R$, i.e. a function such that for $0\not=a\in R$ and $b\in R$ arbitrary either $a\mid b$ or $\delta(c)<\delta(a)$ for some $0\not=c\in(a,b)$.

Proof. Let $\delta$ be a Dedekind-Hasse function, $0\not=I\trianglelefteq R$ an ideal. Chose $0\not=a\in I$ of minimal degree. Clearly, $(a)\le I$. Conversely suppose $b\in I$, $b\notin (a)$. Then there exists $0\not=c\in(a,b)\le I$ of degree less then $a$, a contradiction.

If $R$ is a principal ideal domain, then it is factorial, so the definition $$a\longmapsto\text{Number of prime factors of }a$$ is meaningful. If $0\not=a,b\in R$ and $a\nmid b$, then $\gcd(a,b)$ properly divides $a$, hence has strictly less prime factors. But $\gcd(a,b)\in (a,b)$ by Bézout's Lemma. $\square$

Now it can be shown with some bit of commutative algebra that $\mathbb{R}[X,Y]/(X^2+Y^2+1)$ is a principal ideal domain, see for example here.

Is it possible to give a concrete description of a Dedekind-Hasse function on the ring $R=\mathbb{R}[X,Y]/(X^2+Y^2+1)$?

The most well-known application of the Dedekind-Hasse criterion is probably to some rings of integers of quadratic number fields, e.g. the algebraic norm is a Dedekind-Hasse function on $\mathcal{O}_{\mathbb{Q}(\sqrt{-19})} $.

Are there other applications of the Dedekind-Hasse criterion to find some non-obvious examples of principal ideal domains?

Answer 1

Cohn [Math. Proc. Cambridge Philos. Soc. 64, 251--264 (1968)] called a domain $D$ pre-Bezout if pairs of coprime elements of $D$ are co-maximal (if $x,y\in D$ do not have a common factor then $ux+vy=1$ for some $u,v$). Call a domain $D$ atomic if every nonzero non unit of $D$ is expressible as a finite product of irreducible elements (atoms).

In proving that an atomic pre-Bezout domain is a PID, Hasse's criterion was briefly used in pages 18-20 of [manuscripta math. 35 (1981),1--26].

The "proof" meant:

a) showing that in a pre-Bezout domain an atom is a prime. (Let $x$ be an atom and consider $x|ab$ for $a,b\in D.$ If $x\nshortmid a$ then $x$ and $a$ are coprime and so for some $u,v\in D$ we have $ux+va=1$ and so $uxb+vab=b$ forcing $x|b.)$

b) noting that an atomic domain in which every atom is a prime is a UFD and using this fact to establish that we can define a function $f$ from $% D\backslash \{0\}$ to the set of natural numbers such that $a|b$ implies that $f(a)\leq f(b)$ with equality only if $b|a$ also. (If $% a=p_{1}^{n_{1}}p_{2}^{n_{2}}...p_{r}^{n_{r}}$ then $f$ defined by $f(a)=\sum n_{i}$ would do.)

c) noting that $D$ is a UFD and that if $a,b\in D$ such that $a$ and $b$ don't divide each other and if $d=GCD(a,b)$ then $f(d)<\min (f(a),f(b)$ ) and $GCD(a/d,b/d)=1$ in the pre-Bezout domain $D$ means the existence of $% u,v $ such that $u(a/d)+v(b/d)=1,$ forcing $d\in (a,b).$

Now it does work but I don't know if it should be considered as an application, or an answer to the question: "Are there other applications of the Dedekind-Hasse criterion to find some non-obvious examples of principal ideal domains?". Any guidance will be appreciated.