K1 of Von Neumann regular rings

Menal, Pere; Moncasi, Jaume

doi:10.1016/0022-4049(84)90064-1

Cited by 39 publications

(19 citation statements)

References 11 publications

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“…Results in this direction exist for rings with stable rank one (see, e.g. [25], and [33]). Note that the previous theorem would follow immediately from Corollary 3.7 in case we knew that R/I was separative for any separative QB-ring R. This is true for C * -algebras, as proved in [17,Proposition 3.4], but the corresponding result in the algebraic context remains open.…”

Section: Proposition 34 Let R Be Any Qb-ring (Unital or Not) Denotmentioning

confidence: 97%

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Non-stable $K$-theory for $QB$-rings

Ara¹,

Domènech²

2007

MATH. SCAND.

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We study the class of QB-rings that satisfy the weak cancellation condition of separativity for finitely generated projective modules. This property turns out to be crucial for proving that all (quasi-)invertible matrices over a QB-ring can be diagonalised using row and column operations. The main two consequences of this fact are: (i) The natural map GL 1 (R) → K 1 (R) is surjective, and (ii) the only obstruction to lift invertible elements from a quotient is of K-theoretical nature. We also show that for a reasonably large class of QB-rings that includes the prime ones, separativity always holds.

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Section: Proposition 34 Let R Be Any Qb-ring (Unital or Not) Denotmentioning

confidence: 97%

“…These are known in the case of rings with stable rank one (see, e.g. [25]), separative exchange rings (see [7], [29]) and also extremally rich C * -algebras ( [17]). We know already that these results are relevant for QB-rings as their stable rank is usually different from one.…”

Section: Non-stable K-theorymentioning

confidence: 99%

Non-stable $K$-theory for $QB$-rings

Ara¹,

Domènech²

2007

MATH. SCAND.

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“…In [4] P. Menal and J. Moncasi introduced the concept of unit-1-stable rank . A unimodular pair (al, a2) E U2(A) is said to be totally reducible if there exists an element u E A -1 such that al +ua2 E A-1 .…”

Section: Introductionmentioning

confidence: 99%

On the unit 1-stable rank of rings of analytic functions

Doménech

Sobregrau²,

Brufal³

1992

Publicacions Matemàtiques

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“…The group GLnB/E,(A, B) = K1 (A, B) = GLI B/W(A, B) is also an abelian group which does not depend on n for n >_ 2, where W(A, B) is the subgroup of GLl A generated by the elements of the form (a + c + abc) (a + c + cba) -1 with a E B, b E A, c E 1 + B (see [17], [22]) . So the group G,(A, B)/E(A, B), which classifies all subgroups H of GL,,A normalized by EnA and with the same "leveF B, is two-step nilpotent .…”

Section: Furthermore In This Case H = [E2 a E2a] Is A Normal Subgrmentioning

confidence: 99%