I have two graded valuation rings, $R$ and $S$, that I want to relate by a function $f:R\rightarrow S$. The problem is that $f$ is only a group homomorphism under addition for two elements of $R$ that have the same valuation. In any case, however, if $v_R(u) < v_R(v)$ then $v_R(f(u+v)) = \min\{v_R(u), v_R(v)\}$.
What can I do from here?