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Bass’s first stable range condition. (English) Zbl 0547.16017

In this note much of the basic known information about rings with stable range 1 is summarized, and some new information is added. The note contains, in particular, the contents of a short paper on the first stable range condition circulated by Kaplansky about 1971. Many examples of rings and \(C^*\)-algebras (with and without unit) having stable range 1 are given, the left-right symmetry and Morita-invariance of stable range 1 are discussed, and a cancellation theorem is proved. Namely, over a ring with stable range 1, any finitely generated projective module P may be cancelled from direct sums (that is, \(P\oplus A\cong P\oplus B\) implies \(A\cong B)\). There is a somewhat more general cancellation theorem, not mentioned in this note, due to E. Evans [Pac. J. Math. 46, 115-121 (1973; Zbl 0272.13006); Theorem 2]: any module whose endomorphism ring has stable range 1 may be cancelled from direct sums.
Reviewer: K.R.Goodearl

MSC:

16E20 Grothendieck groups, \(K\)-theory, etc.
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16D40 Free, projective, and flat modules and ideals in associative algebras

Citations:

Zbl 0272.13006
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Full Text: DOI

References:

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