# Multiplicatively monotone norm is constant on associate classes

## Statement

Suppose $R$ is a commutative unital ring and $N$ is a norm on $R$ that is multiplicatively monotone: $N(ab) \ge \max \{ N(a), N(b) \}$ whenever $a,b \in R$ are such that $ab \ne 0$. Then, if $a,b \in R$ are associate elements, we have $N(a) = N(b)$.

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

## Proof

Given: A multiplicatively monotone Euclidean norm $N$ on a commutative unital ring $R$. Two elements $a,b \in R$ that are associate elements.

To prove: $N(a) = N(b)$.

Proof: By definition of associate elements, there exist elements $c,d \in R$ such that $a = bc, b = da$. By the definition of multiplicatively monotone, we get:

$a = bc \implies N(a) \ge \max \{ N(b), N(c) \} \ge N(b)$.

Similarly:

$b = da \implies N(b) \ge N(a)$.