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 fact about::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:

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