# Simultaneous bases for submodules of free modules over PIDs, proof without Smith normal forms?

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!