Stably free modules over $\mathbf{R}[X]$ of rank $> \dim\mathbf{R}$ are free

We prove that for any finite-dimensional ring R and n > dim R+2, the group E n (R[X]) acts transitively on Um n (R[X]). In particular, we obtain that for any finite-dimensional ring R, all finitely generated stably free modules over R[X] of rank > dim R are free. This result was only known for Noetherian rings. The proof we give is short, simple, and constructive.