Stable range condition
In mathematics, particular in abstract algebra and algebraic K-theory, the stable range of a ring [math]\displaystyle{ R }[/math] is the smallest integer [math]\displaystyle{ n }[/math] such that whenever [math]\displaystyle{ v_0,v_1,...,v_n }[/math] in [math]\displaystyle{ R }[/math] generate the unit ideal (they form a unimodular row), there exist some [math]\displaystyle{ t_1,...,t_n }[/math]in [math]\displaystyle{ R }[/math] such that the elements [math]\displaystyle{ v_i - v_0t_i }[/math] for [math]\displaystyle{ 1\le i \le n }[/math] also generate the unit ideal.
If [math]\displaystyle{ R }[/math] is a commutative Noetherian ring of Krull dimension [math]\displaystyle{ d }[/math] , then the stable range of [math]\displaystyle{ R }[/math] is at most [math]\displaystyle{ d+1 }[/math] (a theorem of Bass).
Bass stable range
The Bass stable range condition [math]\displaystyle{ SR_m }[/math] refers to precisely the same notion, but for historical reasons it is indexed differently: a ring [math]\displaystyle{ R }[/math] satisfies [math]\displaystyle{ SR_m }[/math] if for any [math]\displaystyle{ v_1,...,v_m }[/math] in [math]\displaystyle{ R }[/math] generating the unit ideal there exist [math]\displaystyle{ t_2,...,t_m }[/math] in [math]\displaystyle{ R }[/math] such that [math]\displaystyle{ v_i - v_1t_i }[/math] for [math]\displaystyle{ 2\le i \le m }[/math] generate the unit ideal.
Comparing with the above definition, a ring with stable range [math]\displaystyle{ n }[/math] satisfies [math]\displaystyle{ SR_{n+1} }[/math]. In particular, Bass's theorem states that a commutative Noetherian ring of Krull dimension [math]\displaystyle{ d }[/math] satisfies [math]\displaystyle{ SR_{d+2} }[/math]. (For this reason, one often finds hypotheses phrased as "Suppose that [math]\displaystyle{ R }[/math] satisfies Bass's stable range condition [math]\displaystyle{ SR_{d+2} }[/math]...")
Stable range relative to an ideal
Less commonly, one has the notion of the stable range of an ideal [math]\displaystyle{ I }[/math] in a ring [math]\displaystyle{ R }[/math]. The stable range of the pair [math]\displaystyle{ (R,I) }[/math] is the smallest integer [math]\displaystyle{ n }[/math] such that for any elements [math]\displaystyle{ v_0,...,v_n }[/math] in [math]\displaystyle{ R }[/math] that generate the unit ideal and satisfy [math]\displaystyle{ v \equiv 1 }[/math] mod [math]\displaystyle{ I }[/math] and [math]\displaystyle{ v_i \equiv 0 }[/math] mod [math]\displaystyle{ I }[/math] for [math]\displaystyle{ 0\le i \le n-1 }[/math], there exist [math]\displaystyle{ t_1,...,t_n }[/math] in [math]\displaystyle{ R }[/math] such that [math]\displaystyle{ v_i - v_0t_i }[/math] for [math]\displaystyle{ 1\le i \le n }[/math] also generate the unit ideal. As above, in this case we say that [math]\displaystyle{ (R,I) }[/math] satisfies the Bass stable range condition [math]\displaystyle{ SR_{n+1} }[/math].
By definition, the stable range of [math]\displaystyle{ (R,I) }[/math] is always less than or equal to the stable range of [math]\displaystyle{ R }[/math].
References
- H. Chen, Rings Related Stable Range Conditions, Series in Algebra 11, World Scientific, Hackensack, NJ, 2011. [1]
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