Suppose $R \neq 0$ is a commutative ring with $1$. The following is well known:
(Isomorphism Theorem for Finitely Generated Free Modules) [FGFM] $R^{n}\cong R^{m}$ as $R$-modules if and only if $n=m$.
One proof of this result (if I recall correctly) was somehow based on the idea that we can quotient out by a maximal ideal of $R$ (whose existence is guaranteed by Zorn's Lemma), and reduce the situation to that of vector spaces. However, I am interested in different approach. Recall
(Nakayama's Lemma) Suppose $M$ is a finitely generated $R$-module, and $M=IM$ where $I$ is an ideal contained in Jacobson radical of $R$. Then, $M=0$.
My question is:
Can we prove FGFM using Nakayama's Lemma?
My main motivation in asking this question is two-fold:
1) to see the power and usefulness of Nakayama's Lemma, and
2) to see a nice and short proof of FGFM that does not involve reducing the problem to a result from linear algebra.
I appreciate any input :)