When do we have equality in the third axiom of valuations?

Question

Let $R$ be a ring and $v: R \to \mathbb Z \cup \{+\infty\}$ a map that meets following axioms:

1. $v(a) = +\infty \iff a=0$
2. $v(ab) = v(a)+v(b)$
3. $v(a+b) \geq \min\{v(a),v(b)\}$

I have to show that $v(a) \neq v(b)$ implies $v(a+b) = \min\{v(a),v(b)\}$, but I have no idea where to begin, can anyone give me some hints?

Answer 1

Assume $v(a)>v(b)$ and $v(a+b)>v(b)$. Then $$ v(b)=v((a+b)+(-a))\ge\min\{v(a+b),v(-a)\}=\min\{v(a+b),v(a)\}>v(b).$$