# Dedekind-Hasse norm

From Commalg

## Statement

A **Dedekind-Hasse norm** on a commutative unital ring is a function from the nonzero elements of to the set of nonnegative integers, satisfying the following condition:

Whenever are both nonzero, then one of these cases holds:

- is an element of the ideal . In other words, .
- There is a nonzero element in the ideal whose norm is strictly smaller than that of .

## Relation with other properties

### Stronger properties

- Euclidean norm:
*For proof of the implication, refer Euclidean implies Dedekind-Hasse and for proof of its strictness (i.e. the reverse implication being false) refer Dedekind-Hasse not implies Euclidean* - Multiplicative Dedekind-Hasse norm
- Multiplicative Euclidean norm

## Facts

- A commutative unital ring that admits a Dedekind-Hasse norm is a principal ideal ring.
*For full proof, refer: Dedekind-Hasse norm implies principal ideal ring*