# Multiplicatively monotone Euclidean norm

From Commalg

*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 is a Euclidean norm on a commutative unital ring , we say that is multiplicatively monotone if for any such that :

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## 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 for in an integral domain with a multiplicatively monotone Euclidean norm, then there is no pair with , and .
*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*