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'.