Multiplicatively monotone norm is constant on associate classes

Statement

Suppose $R$ is a commutative unital ring and $N$ is a Euclidean norm on $R$ that is multiplicatively monotone: $N(ab)\geq \max\{N(a),N(b)\}$ whenever $a,b\in R$ are such that $ab\neq 0$. 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:

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

Similarly:

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