Multiplicatively monotone Euclidean norm

This article defines a property that can be evaluated for a Euclidean norm on a commutative unital ring

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 $ab\neq 0$:

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

Relation with other properties

Stronger properties

• 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. For full proof, refer: Multiplicatively monotone Euclidean norm is constant on associate classes
• If $b|a$ for $b\neq 0$ in an integral domain with a multiplicatively monotone Euclidean norm, then there is no pair $(q,r)$ with $a=bq+r$, $r\neq 0$ and $N(r)<N(b)$. For full proof, refer: Multiplicatively monotone Euclidean norm admits unique Euclidean division for exact divisor
• A Euclidean norm that is both filtrative and multiplicatively monotone is a uniquely Euclidean norm. For full proof, refer: filtrative and multiplicatively monotone implies uniquely Euclidean