Multiplicatively monotone Euclidean norm

Definition
A Euclidean norm is termed multiplicatively monotone if the norm of a nonzero product of two elements is at least equal to the norms of the elements. In symbols, if $$N$$ is a Euclidean norm on a commutative unital ring $$R$$, we say that $$N$$ is multiplicatively monotone if for any $$a,b \in R$$ such that $$ab \ne 0$$:

$$N(ab) \ge \max \{ N(a), N(b) \}$$.

Stronger properties

 * Weaker than::Multiplication-additive Euclidean norm: Here, the norm of a product equals the sum of the norms.
 * Multiplicative Euclidean norm as long as there are no elements of norm zero.

Facts

 * A multiplicatively monotone Euclidean norm takes the same value on associate elements.
 * If $$b|a$$ for $$b \ne 0$$ in an integral domain with a multiplicatively monotone Euclidean norm, then there is no pair $$(q,r)$$ with $$a = bq + r$$, $$r \ne 0$$ and $$N(r) < N(b)$$.
 * A Euclidean norm that is both filtrative and multiplicatively monotone is a uniquely Euclidean norm.