Multiplicatively monotone norm

Definition
A multiplicatively monotone norm on a commutative unital ring is a function from its nonzero elements to the nonnegative integers with the property that the norm of a product is at least equal to the norms of the factors.

In symbols, it is a function $$N: R \setminus \{ 0 \} \to \mathbb{N}_0$$ such that for $$ab \ne 0$$, we have:

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

This definition is typically used for integral domains.

Stronger properties

 * Weaker than::Strictly multiplicatively monotone norm

Facts

 * Multiplicative and positive implies multiplicatively monotone
 * Filtrative and multiplicatively monotone Euclidean implies uniquely Euclidean
 * Multiplicatively monotone Euclidean norm admits unique Euclidean division for exact divisor