Dedekind-Hasse norm

Statement

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

Whenever $a,b\in R$ are both nonzero, then one of these cases holds:

• $a$ is an element of the ideal $(b)$. In other words, $b|a$.
• There is a nonzero element in the ideal $(a,b)$ whose norm is strictly smaller than that of $b$.

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