# Multiplicatively monotone Euclidean norm admits unique Euclidean division for exact divisor

## Statement

Suppose $R$ is an commutative unital ring with a Euclidean norm $N$ that is multiplicatively monotone: $N(ab) \ge \max \{ N(a), N(b) \}$ whenever $ab \ne 0$.

Suppose $a = bc$ for $b \ne 0$. Then, we cannot write:

$a = bq + r$

with $r \ne 0$ and $N(r) < N(b)$.

## Proof

Given: An integral domain $R$ with multiplicatively monotone Euclidean norm $N$. $b \ne 0$ and $a = bc$.

To prove: We cannot write $a = bq + r$ with $N(r) < N(b)$.

Proof: Suppose $a = bq + r$ with $N(r) < N(b)$. Then, since $a = bc$, we get:

$bc = bq + r \implies b(c - q) = r$.

Take $N$ both sides, to get:

$N(b(c-q)) = N(r)$.

Since $N$ is multiplicatively monotone, we get:

$N(b(c-q)) \ge \max \{ N(b), N(c - q) \}$.

On the other hand, we have, by assumption:

$N(r) < N(b)$.

This is a contradiction, completing the proof.