forgetful functor from RMod to Ab as conservative functor

The forgetful functor from RMod to Ab is conservative

Suppose \(\mathrm{Forget}(f: M \rightarrow N)\) is a group-isomorphism, then \(f\) is bijective and r-linear. Let \(f^{-1}: N \rightarrow M\) be the inverse morphism.

Then it remains to show, that \(f^{-1}\) is R-linear: Suppose \(r \in R, n \in N\), then

\begin{align*} f(f^{-1}(rn)) =& rn \\ =& r \cdot (f \circ f^{-1})(n) \\ =& f(r \cdot f^{-1}(n)) \end{align*}

Hence injectivity of \(f\) implies

\begin{align*} r \cdot f^{-1}(n) =& f^{-1}(rn) \end{align*}