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"0"t""i" 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 (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"1"t""i" for 2 ≤ "i" ≤ "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"n 1 mod "I" and "v"i 0 mod "I" for 0 ≤ "i" ≤ "n"-1, there exist "t"1, ... , "t""n" in "R" such that "v""i" - "v"0"t""i" for 1 ≤ "i" ≤ "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".
References.
H. Chen, Rings Related Stable Range Conditions, Series in Algebra 11, World Scientific,
Hackensack, NJ, 2011. 