Theorem 1.6. If $F$ is a free abelian group of finite rank $n$ and $G$ is a nonzero subgroup of $F$, then there exists a basis $\{x_1,\ldots,x_n\}$ of $F$, an integer $r$ ($1\leq r\leq n$) and positive integers $d_i,\ldots,d_r$, such that $d_1\mid d_2\mid\cdots\mid d_r$, and $G$ is free abelian with basis $\{d_1x_1,\ldots,d_rx_r\}$.
If $R$ is a PID, $F$ is a free $R$-module of finite rank $n$ and $M$ is a nonzero submodule of $F$, then there exists a basis $\{x_1,\ldots,x_n\}$ of $F$, an integer $r$ ($1\leq r\leq n$) and elements $d_i,\ldots,d_r\in R$, such that $d_1\mid d_2\mid\cdots\mid d_r$, and $M$ is free with basis $\{d_1x_1,\ldots,d_rx_r\}$?