stably free module
Some projective modules are not free, yet become free the moment you add a few free directions to them — like an awkwardly shaped region that tiles space perfectly once combined with a copy of itself. A stably free module is exactly this: not free on its own, but free after the harmless addition of a free summand. Such modules are the reason K_0 reads stable isomorphism rather than honest isomorphism, and they are the first concrete sign that projective does not imply free.
A right R-module P is stably free if there exist nonnegative integers m and n with P ⊕ R^m isomorphic to R^n. Equivalently, P is finitely generated projective and its class [P] in the reduced group K_0-tilde(R) is zero; that is, [P] lies in the image of the free modules. Every stably free module is projective, and every free module is stably free, but neither converse holds in general.
The classic source of genuinely nonfree stably free modules is the tangent bundle of even spheres, algebraized: over the real coordinate ring R = R[x_0, ..., x_n]/(x_0^2 + ... + x_n^2 - 1) of the n-sphere with n even, the module of tangent vector fields is stably free (it is a summand complementing the normal line in the trivial rank-(n+1) bundle) but not free, by the hairy ball theorem. So whether stably free implies free is a substantive question, controlled by the ring's geometry and detected by K_0.
Over the real coordinate ring of the 2-sphere, the kernel P of the surjection R^3 -> R sending (f, g, h) to x f + y g + z h is stably free: P ⊕ R is isomorphic to R^3. But P is not free, because a free generator would be a nonvanishing tangent field, which the hairy ball theorem forbids on S^2.
The tangent module of S^2 is stably free but not free.
Over a ring R, every stably free module is free iff R has the cancellation property in the relevant rank range. For commutative Noetherian rings of Krull dimension d, Bass's cancellation theorem guarantees that stably free modules of rank greater than d are free, so failures occur only in low rank relative to dimension.