In Advanced Linear Algebra by S. Roman, the author cites the following theorem without proof.

Theorem 6.7 Let $M$ be a free $R$-module of rank $n$, where $R$ is a principal ideal domain. Let $N$ be a submodule of $M$ that is free of rank $k\leq n$. Then there is a basis $\mathcal{B}$ for $M$ that contains a subset $S=\{v_1,\ldots,v_k\}$ for which $\{r_1v_1,\ldots,r_kv_k\}$ is a basis for $N$, for some nonzero elements $r_1,\ldots,r_k$ of $R$.

As far as I can see it, this cannot be deduced directly from the usual structure theorem for finitely generated modules over PIDs. However, here Smith normal form does the job. Also I know that . So I wonder whether there is any other proofs for this fact for general PIDs? Especially one that is not such "matrix-theoretic"? Thanks for any references!


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