This is the theorem the title refers to. In his Basic Algebra I, Jacobson proves it by means of a Lemma:
Lemma Let $D$ be a PID and $K$ be a submodule of $D^{(n)}$ (the free module of rank $n$). Then
$K$ is finitely generated;
$K$ is free of rank $\le n$.
Question What is the relevance of conclusion 2. to the subsequent proof? I guess it is not needed.
Indeed, Jacobson's proof goes as follows. Take a finitely generated module $M$ over the PID $D$ and a generating homomorphism $ \eta\colon D^{(n)} \to M$. Then the kernel $K$ of $\eta$ is finitely generated and we have a relations matrix $A$, whose rows are a set of generators for $K$. We then apply the machinery of Smith normal form to $A$.
It seems to me that we made no use of 2. This point guarantees that we can take $A$ of full rank, but that is something we don't need. Am I wrong?
Thank you.