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
- 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