Alternate characterization of rank

Let $R$ be an integral domain and $M$ be a $R$ -module.

Suppose that $M$ has rank $n$ and $S = {m_{1}, \dots, m_{n}}$ is a maximal linearly independent set of elements in $M$ . Let $N$ be the submodule generated by $S$ . Prove that $N ≃ R^{n}$ and $M / N$ is a torsion $R$ -module.
Conversely, prove that if $M$ contains a submodule $N$ that is free of rank $n$ such that the quotient $M / N$ is a torsion $R$ -module, then $M$ has rank $n$ .

Hints

Show that for any $m \in M$ there is a nonzero $r \in R$ such that $r m = r_{1} m_{1} + \dots + r_{n} m_{n}$ for some $r_{i} \in R$ .
Let ${m_{1}, \dots, m_{n + 1}}$ be a set of $n + 1$ elements of $M$ . Find some nonzero $r_{i} \in R$ so that $r_{i} m_{i}$ can be written using a basis for $N$ . Then show the $r_{i} m_{i}$ (and hence $m_{i}$ ) are linearly dependent.