Is the inclusion map always a module homomorphism?

Question

Suppose $R$ is some commutative ring and $G$ a finite group so that $R[G]$ is the usual group ring. If $M$ is some $R[G]$-module, then we can inject $M\hookrightarrow M\oplus R[G]^n$ for some $n\geq 1$. Is this necessarily an $R[G]$-homomorphism?

Answer 1

Yes, the inclusion of a factor into a direct product $M \to M \oplus N$ defined by $m \mapsto (m, 0)$ is always an injective homomorphism no matter what modules you choose for $M$ and $N$. In particular, you can inject $M \to M \oplus R[G]^n$ not just for some $n$, but in fact for any $n$.

If $M$ and $N$ are $G$-modules then it would be a good exercise for you to write out the definition of the $G$-module action on $M \oplus N$ and the axioms that a homomorphism must satisfy, and then check that the inclusion map $M \to M \oplus N$ does indeed satisfy these axioms.