nLab Dedekind-Hasse norm

Idea

The notion of a Dedekind-Hasse norm is a generalization of the notion of the degree function in a Euclidean domain. It makes clear the links between a principal ideal domain and an Euclidean domain: a pid is like an Euclidean domain but with a weaker notion of Euclidean division.

Definition and characterization of pids

Definition

Let $R$ be an integral domain. A Dededekind-Hasse norm is a function $v \colon R \rightarrow \mathbb{N}$ such that:

• $v(s)=0 \Leftrightarrow s=0$

• $\forall a \in R$, $\forall (b \neq 0) \in R$:

  • $b|a$, or

  • $\exists p,q,r \in R$ such that $pa = bq + r$ and $0 \lt v(r) \lt v(b)$

Proposition

Let $R$ be an integral domain. Then it is a principal ideal domain iff it possesses a Dedekind-Hasse norm.

Proof

Suppose that $R$ possesses a Dedekind-Hasse norm. Let $I$ be a non-zero ideal. Let $b$ be a non-zero element of $I$ of minimal norm. We know that $(b) \subseteq I$. Let $a$ be an element of $I$. Suppose that $b$ doesn’t divide $a$. Then, there exists $p,q,r \in R$ such that $pa = bq+r$ and $0 \lt v(r) \lt v(b)$. Thus, $r=pa-bq \in I$, $r \neq 0$ and $v(r) \lt v(b)$, absurd! Therefore $b|a$ and $a \in (b)$. Thus, $I \subseteq (b)$ and $I = (b)$. We have proved that $R$ is a pid.

Suppose that $R$ is a pid. Thus, it is a UFD. Put $v(s)=0$ if $s=0$, and $v(s)$ equal to $2^{n}$ where $n$ is the number of irreducible elements in the factorization of $n$, if $s \neq 0$. Let $a \in R$ and $(b \neq 0) \in R$. Suppose that $b$ doesn’t divide $a$. We know that $(a,b) = (r)$ and thus there exists $p,q,r \in R$ such that $pa+bq = r$. $r$ divides $b$ but $b$ doesn’t divide $r$ because it would imply that $b$ divides $a$. Thus, there is strictly less irreducible elements in the factorization of $r$ than in the one of $b$ and $v(r) \lt v(b)$. Moreover $r \neq 0$ because $(a,b) = (r)$ and $b \neq 0$. Thus $0 \lt v(r) \lt v(b)$. We have proved that $v$ is a Dedekind-Hasse norm.

In constructive mathematics

In constructive mathematics, there are different types of integral domains, yielding different types of Dedekind-Hasse norms. For example, if the integral domain has a tight apartness relation $a # b$, such as in a Heyting integral domain, then one can use the tight apartness relation instead of denial inequality in the second condition

• $v(s)=0 \Leftrightarrow s=0$

• $\forall a \in R$, $\forall (b \# 0) \in R$:

  • $b|a$, or

  • $\exists p,q,r \in R$ such that $pa = bq + r$ and $0 \lt v(r) \lt v(b)$

In constructive mathematics, integral domains with Dedekind-Hasse norms are not the same as principal ideal domains in constructive mathematics: the integers is a discrete integral domain with a Dedekind-Hasse norm because it is a Euclidean domain, but is not a principal ideal domain unless excluded middle holds.

It remains to see if integral domains with Dedekind-Hasse norms are still the same as Bézout unique factorization domains or Noetherian Bézout domains in constructive mathematics.

References

Named after Helmut Hasse.

• Wikipedia, Dedekind-Hasse norm