Let $R$ be a ring and $m$ be a maximal ideal.
$m R_m$ denotes the unique maximal ideal of the localization $R_m$ of $R$ at $m$. Let $A$ be an $R$-module. The localization of $A$ at $m$ is denoted by $A_m$.
I have to prove that the map $\pi : A \rightarrow A_m, a \mapsto \frac{a}{1}$ induces an isomorphism $A/mA \cong A_m/mR_m A_m$. (*)
My approach : I have shown that $\pi : R \rightarrow R_m, r \mapsto \frac{r}{1}$ induces an isomorphism $R/mR \cong R_m / mR_m$. Since the notion of localization of a ring can be generalized to the localization of a module, I think I can directly conclude that (*) holds.
Is my approach correct or is there still something to prove ?