Stable range condition

In mathematics, particular in abstract algebra and algebraic K-theory, the stable range of a ring is the smallest integer such that whenever in generate the unit ideal (they form a unimodular row), there exist some in such that the elements for also generate the unit ideal.

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

Bass stable range

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The Bass stable range condition   refers to precisely the same notion, but for historical reasons it is indexed differently: a ring   satisfies   if for any   in   generating the unit ideal there exist   in   such that   for   generate the unit ideal.

Comparing with the above definition, a ring with stable range   satisfies  . In particular, Bass's theorem states that a commutative Noetherian ring of Krull dimension   satisfies  . (For this reason, one often finds hypotheses phrased as "Suppose that   satisfies Bass's stable range condition  ...")

Stable range relative to an ideal

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Less commonly, one has the notion of the stable range of an ideal   in a ring  . The stable range of the pair   is the smallest integer   such that for any elements   in   that generate the unit ideal and satisfy   mod   and   mod   for  , there exist   in   such that   for   also generate the unit ideal. As above, in this case we say that   satisfies the Bass stable range condition  .

By definition, the stable range of   is always less than or equal to the stable range of  .

References

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  • H. Chen, Rings Related Stable Range Conditions, Series in Algebra 11, World Scientific, Hackensack, NJ, 2011. [1]
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