Bass stable range

Remarks

If $\mathrm{sr}(R) \le n$, then $\mathrm{sr}(R) \le n'$ for every $n' \ge n$. Bass's original notation $SR_{n+1}$ is the condition $\mathrm{sr}(R) \le n$ here, so the index is shifted by one. The condition $\mathrm{sr}(R) \le 1$ says: if $Ra + Rb = R$, then $a + tb$ is a unit for some $t \in R$. Bass proved that a commutative Noetherian ring of Krull dimension $d$ has $\mathrm{sr}(R) \le d + 1$.

Example: a field $F$ has $\mathrm{sr}(F) = 1$. If $Fa + Fb = F$ and $a \ne 0$, take $t = 0$; if $a = 0$, then $b \ne 0$ and $a + 1 \cdot b = b$ is a unit. Every commutative local ring also has stable rank $1$.

Non-example of $\mathrm{sr} \le 1$: the ring $\Z$. The row $(2, 5)$ is unimodular since $3 \cdot 2 - 5 = 1$, but $2 + 5t$ is never $\pm 1$ for an integer $t$. So $\mathrm{sr}(\Z) \ge 2$, and Bass's bound with $d = 1$ gives $\mathrm{sr}(\Z) = 2$.