Greetings,
in my studies I went into a statement "minimal generating set of a free module over a local ring is a free basis". The statement came without a proof, just with a reference to Kaplansky's theorem. I was unsuccessful trying to prove the statement myself, and I couldn't find the proof elsewhere either. I would be grateful for any hint.



Minimal generating set of a free module over local ring
Any free module $M$ is projective. Then, when $M$ is finitely generated, you can use the argument given in [Matsumura, Commutative Ring Theory](http://books.google.it/books?id=yJwNrABugDEC&printsec=frontcover&dq=matsumura+commutative+ring+theory&hl=en&sa=X&ei=zEHqTtrqLcrS4QSFtdTsCA&redir_esc=y#v=onepage&q=kaplansky&f=false), proof of Theorem 2.5 page 10.


Matsumura also expains the argument used by Kaplanski ["Projective modules", Ann. Math. **68** (1958)] in the general case.


