# Quickest route to the structure theorem for finitely generated modules over a PID

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

1. $K$ is finitely generated;

2. $K$ 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.

• The title doesn't fit to the question. – Martin Brandenburg May 9 '13 at 17:57

.. A first result we shall need is that $K$ is finitely generated. This will follow from the following stronger result.
And indeed, in Jacobson's presentation 2.) is not needed to prove the structure theorem, we don't have to bother with finding a base for $K$. Of course, it is still a nice to know and proving it does not require that much work.