# Multiplicatively monotone norm is constant on associate classes

From Commalg

## Statement

Suppose is a commutative unital ring and is a norm on that is multiplicatively monotone: whenever are such that . Then, if are associate elements, we have .

Note that the proof does not in fact use the fact that the norm is Euclidean.

## Proof

**Given**: A multiplicatively monotone Euclidean norm on a commutative unital ring . Two elements that are associate elements.

**To prove**: .

**Proof**: By definition of associate elements, there exist elements such that . By the definition of multiplicatively monotone, we get:

.

Similarly:

.