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.
Any free module $M$ is projective. Then, when $M$ is finitely generated, you can use the argument given in Matsumura, Commutative Ring Theory, 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.