Stable range condition

In mathematics, particular in abstract algebra and algebraic K-theory, the stable range of a ring $$R$$ is the smallest integer $$n$$ such that whenever $$v_0,v_1,...,v_n$$ in $$R$$ generate the unit ideal (they form a unimodular row), there exist some $$t_1,...,t_n$$in $$R$$ such that the elements $$v_i - v_0t_i$$ for $$1\le i \le n$$ also generate the unit ideal.

If $$R$$ is a commutative Noetherian ring of Krull dimension $$d$$, then the stable range of $$R$$ is at most $$d+1$$ (a theorem of Bass).

Bass stable range
The Bass stable range condition $$SR_m$$ refers to precisely the same notion, but for historical reasons it is indexed differently: a ring $$R$$ satisfies $$SR_m$$ if for any $$v_1,...,v_m$$ in $$R$$ generating the unit ideal there exist $$t_2,...,t_m$$ in $$R$$ such that $$v_i - v_1t_i$$ for $$2\le i \le m$$ generate the unit ideal.

Comparing with the above definition, a ring with stable range $$n$$ satisfies $$SR_{n+1}$$. In particular, Bass's theorem states that a commutative Noetherian ring of Krull dimension $$d$$ satisfies $$SR_{d+2}$$. (For this reason, one often finds hypotheses phrased as "Suppose that $$R$$ satisfies Bass's stable range condition $$SR_{d+2}$$...")

Stable range relative to an ideal
Less commonly, one has the notion of the stable range of an ideal $$I$$ in a ring $$R$$. The stable range of the pair $$(R,I)$$ is the smallest integer $$n$$ such that for any elements $$v_0,...,v_n$$ in $$R$$ that generate the unit ideal and satisfy $$v \equiv 1$$ mod $$I$$ and $$v_i \equiv 0$$ mod $$I$$ for $$0\le i \le n-1$$, there exist $$t_1,...,t_n$$ in $$R$$ such that $$v_i - v_0t_i$$ for $$1\le i \le n$$ also generate the unit ideal. As above, in this case we say that $$(R,I)$$ satisfies the Bass stable range condition $$SR_{n+1}$$.

By definition, the stable range of $$(R,I)$$ is always less than or equal to the stable range of $$R$$.