In mathematics, particular in abstract algebra and algebraic K-theory, the stable range of a ring R {\displaystyle R} is the smallest integer n {\displaystyle n} such that whenever v 0 , v 1 , . . . , v n {\displaystyle v_{0},v_{1},...,v_{n}} in R {\displaystyle R} generate the unit ideal (they form a unimodular row), there exist some t 1 , . . . , t n {\displaystyle t_{1},...,t_{n}} in R {\displaystyle R} such that the elements v i − v 0 t i {\displaystyle v_{i}-v_{0}t_{i}} for 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} also generate the unit ideal. If R {\displaystyle R} is a commutative Noetherian ring of Krull dimension d {\displaystyle d} , then the stable range of R {\displaystyle R} is at most d + 1 {\displaystyle d+1} (a theorem of Bass).
Bass stable range The Bass stable range condition S R m {\displaystyle SR_{m}} refers to precisely the same notion, but for historical reasons it is indexed differently: a ring R {\displaystyle R} satisfies S R m {\displaystyle SR_{m}} if for any v 1 , . . . , v m {\displaystyle v_{1},...,v_{m}} in R {\displaystyle R} generating the unit ideal there exist t 2 , . . . , t m {\displaystyle t_{2},...,t_{m}} in R {\displaystyle R} such that v i − v 1 t i {\displaystyle v_{i}-v_{1}t_{i}} for 2 ≤ i ≤ m {\displaystyle 2\leq i\leq m} generate the unit ideal. Comparing with the above definition, a ring with stable range n {\displaystyle n} satisfies S R n + 1 {\displaystyle SR_{n+1}} . In particular, Bass's theorem states that a commutative Noetherian ring of Krull dimension d {\displaystyle d} satisfies S R d + 2 {\displaystyle SR_{d+2}} . (For this reason, one often finds hypotheses phrased as "Suppose that R {\displaystyle R} satisfies Bass's stable range condition S R d + 2 {\displaystyle SR_{d+2}} ...")
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