I'm looking for clarification on Fraleigh's "A First Course in Abstract Algebra" Theorem 38.11.
It states: "Let $G$ be a nonzero free abelian group of finite rank $n$, and let $K$ be a nonzero group of $G$. Then $K$ is free abelian of rank $s \le n$. Furthermore, there exists a basis $\{x_1, x_2, ..., x_n \}$ for $G$ and positive integers $d_1, d_2, ..., d_n$, where $d_i$ divides $d_{i+1}$ for $i = 1, 2, ..., s-1$ such that $\{d_1x_1, d_2x_2, ..., d_sx_s \}$ is a basis for $K$."
What I an confused about is his particular proof, where he determines the basis for $K$ utilizing the canonical division algorithm. He does this without use of modules and I am looking for an alternate proof/clarification of the proof that does not use modules.