Stably free module not finitely generated is free Hi. I have read that stably free modules not finitely generated are free; this is proved in 
M.R. Gabel, stably free projectives over commutative rings, Thesis, Brandeis Univ., Waltham, MA 1972.
But I can't find a proof of that, can anyone of you help me?
 A: Let $P\oplus R^k$ be free.  By induction it's enough to take $k=1$, so the assertion comes down to saying that an infinitely long unimodular row can be completed to an infinite invertible matrix.  But an infinitely long unimodular row contains (after rearrangement) a finitely long unimodular sub-row.  So it suffices to show that any unimodular row with a non-trivial initial unimodular sub-row can be completed.  
Write the row as (s,x,t) where s is is unimodular and x has length 1.  Let p be a column such that s.p = 1.  Consider the matrix
$$\matrix{s&x&t\cr I&p&0\cr 0&0&I\cr}$$
where the $I$'s and $0$'s are identity and 0 matrices of approrpiate sizes.  This is easily seen to be invertible.
A: The proper definition of 'stably free' is as follows: let $\Lambda$ be a ring. Then a $\Lambda$-module $S$ is stably free when $S\oplus\Lambda^n$ is a free module of unspecified rank, finite or infinite, where $n$ is an integer. The reason that $n$ must be finite is to avoid Eilenberg's trick which shows that $P\oplus \Lambda^\infty \cong \Lambda^\infty$ for any countably generated projective module $P$. Here $\Lambda^\infty$ denotes the direct limit ${\lim}_{\to}\Lambda^n$ where $\Lambda^n\subset\Lambda^n\oplus\Lambda\cong\Lambda^{n+1}$ under the inclusion $x\mapsto (x,0)$. 
Gabel's theorem actually holds for non-commutative rings also. There is a proof in T.Y. Lam, Serre's conjecture. Although Gabel's theorem does not hold for projective modules,  a theorem of Kaplansky [Projective modules, Ann. of math, 68:2 (1958) 372 - 377]  says that every projective module is a direct sum of countably generated projective modules, and so even infinitely generated projectives cannot be 'too bad'.
A: I think that the proof contained in http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/stablyfree.pdf (link given by darij grinberg) it's the best proof. Thanks to all
