Strictly multiplicatively monotone norm

Definition
A strictly multiplicatively monotone norm on a commutative unital ring $$R$$ is a function $$N: R \setminus \{ 0 \} \to \mathbb{N}_0$$ such that:


 * For $$ab \ne 0$$, $$N(ab) \ge \max \{ N(a), N(b) \}$$.
 * For $$ab \ne 0$$, $$N(ab) = N(a)$$ if and only if $$a$$ and $$ab$$ are defining ingredient::associate elements.

Facts

 * Strictly multiplicatively monotone norm on Bezout domain is a Dedekind-Hasse norm