I am looking for a possibly general class of algebraic structures (maybe special topological rings) in which one can deduce identities of concrete power series from formal ones. This class should especially contain $\mathbb{R}$ and $\mathbb{C}$.

If e.g. $f \circ g = h$ holds for formal power series $f, g, h \in R[[X]]$, I want to be able to conclude $f(g(r)) = h(r)$ for all $r \in R$ for which both sides of the equation converges in $R$.

For those structures $R$ it is obviously necessary that the Cauchy product of two convergent series (if it converges) equals the ordinary product. I know that this always is the case for series over the complex numbers, but does this also remain true in a more abstract setting?