I'm sort of stuck on this problem and I could use a hint:
Let $R$ be a Euclidean domain with a Euclidean function $d$ such that, for all non-zero $a$ and $b$,
- $d(a) \leq d(ab),$
- $d(a+b) \leq \max\{d(a), d(b)\}.$
For $a, b$ non-zero, prove that $d(a) = d(ab)$ if and only if $b$ is invertible.
I've proven that if $b$ is invertible then $d(a) = d(ab)$, but I'm not sure how to prove the converse. Do I start by assuming that $d(ab) = d(a)$ and then prove that $b$ must have an inverse?