Projective modules over a semi-local ring

Question

I need a little bit of help, I found that theorem, but the book doesn't prove it and gives a reference to another book that I don't have; does anyone have an idea?

Let $R$ be a semi-local ring, and $M$ a finite projective $R$-module. Show that $M$ is free if the localizations $M_m$ have the same rank for all maximal ideals $m$ of $R$.

Answer 1

See Lemma 1.4.4 in Bruns and Herzog, Cohen-Macaulay Rings.