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 $$b$$.

Stronger properties

 * Weaker than::Euclidean norm:
 * Weaker than::Multiplicative Dedekind-Hasse norm
 * Weaker than::Multiplicative Euclidean norm

Facts

 * A commutative unital ring that admits a Dedekind-Hasse norm is a principal ideal ring.