Modules over a group presented via a free group.

Question

Say $G$ is presented via a free group $F$ freely generated by $S=\{s_i, 1=1,2,\dots\}$. Then $\pi:F \rightarrow G$ the canonical projection.

Let $R$ be any commutative ring. Can we follow that any $G$-leftmodule $M$ ist a $F$-leftmodule? By $G$-leftmodule, I mean any abelian group $(M,+)$ with $G$-leftaction, that ist left-compatibel with $+$.

Answer 1

Yes, certainly. Moreover, if $f:P\to Q$ is a homomorphism of rings (in your case $P=RF,Q=RG$), then every $Q$-module $M$ is transformed into a $P$-module by setting $pa=f(p)a$ for $p\in P, a\in M$.