Bass stable range

Summary

The stable range condition of Hyman Bass is a condition on a ring that lets one shorten unimodular rows by elementary operations. The least $n$ for which it holds is the stable rank of the ring. It controls stability results in K-theory. Above the stable rank, projective modules can be cancelled, as in the Bass cancellation theorem. It also gives surjective stability for $K_1$: every element of $K_1(R)$ comes from $GL_n(R)$ once $n$ is at least the stable rank. A commutative Noetherian ring of Krull dimension $d$ has stable rank at most $d + 1$.