Rank and quotients

Let $R$ be an integral domain, $M$ an $R$ -module, and $N$ a submodule of $M$ . Prove that the rank of $M$ is the sum of the ranks of $N$ and $M / N$ .

(You may assume $M$ has finite rank.)

Hints

For part (2), let ${m_{1} + N, \dots, m_{s} + N}$ be a maximal $R$ -linearly independent set in $M / N$ and ${n_{1}, \dots, n_{l}}$ be a maximal $R$ -linearly independent set in $N$ . Show that the set ${m_{1}, \dots, m_{s}, n_{1}, \dots, n_{l}}$ is an $R$ -linearly independent set in $M$ . Then use an alternate characterization of rank.