Let $R$ be a ring and $M$ be a $R$-module. In Example 5.10 of Algebra: Chapter 0 that for $r\in R$, \begin{equation} rM=\{r\cdot m:m\in M\} \end{equation} is a submodule of $M$.
I have no problem showing that this is a subgroup of $M$, but why is it closed under the actions of $R$? That is, given $m\in M$ and $s\in R$, how can we find $n\in M$ such that \begin{equation} s\cdot (r\cdot m)=r\cdot n. \end{equation}
Thanks!