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Stable range condition

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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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