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 v0,v1,..., vn in R generate the unit ideal, there exist some t1,..., tn in R such that the elements vi - v0ti for 1 ≤ i ≤ 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.

Bass stable range

The Bass stable range condition SRm refers to precisely the same notion, but for historical reasons it is indexed differently: a ring R satisfies SRm if for any v1,..., vm in R generating the unit ideal there exist t2,..., tm in R such that vi - v1ti for 2 ≤ i ≤ m generate the unit ideal.
Comparing with the above definition, a ring with stable range n satisfies SRn+1. In particular, Bass's theorem states that a commutative Noetherian ring of Krull dimension d satisfies SRd+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 is the smallest integer n such that for any elements v0,..., vn in R that generate the unit ideal and satisfy vn 1 mod I and vi 0 mod I for 0 ≤ i ≤ n-1, there exist t1,..., tn in R such that vi - v0ti for 1 ≤ i ≤ n also generate the unit ideal. As above, in this case we say that satisfies the Bass stable range condition SRn+1.
By definition, the stable range of is always less than or equal to the stable range of R.