K<sub>1</sub>of separative exchange rings and C*-algebras with real rank zero

Ara, Pere; Goodearl, K. R.; O’Meara, K. C.; Raphael, Robert M.

doi:10.2140/pjm.2000.195.261

Cited by 32 publications

(24 citation statements)

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“…In particular, we see that a large subset of the set of von Neumann regular elements can be understood. This parallels the corresponding result established in [7] (see also [6]), where it is shown that all von Neumann regular matrices over a separative exchange ring can be diagonalised with elementary operations. It also recovers the corresponding result for extremally rich C * -algebras ( [17]), although our methods are different.…”

Section: Diagonalization Of Matrices and Cancellation Conditionssupporting

confidence: 84%

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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: Diagonalization Of Matrices and Cancellation Conditionssupporting

confidence: 84%

“…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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“…Central to this is a common bond shared by exchange rings and regular rings; namely, direct sums of their finitely generated projective modules have the common refinement property. See [2,3,4,15,16] for some details and further references.…”

Section: Introductionmentioning

confidence: 99%

“…is still open, despite a strong consensus that non-separative exchange rings should exist. See [3,4,7] for more background on this problem.…”

Section: Introductionmentioning

confidence: 99%

Gromov translation algebras over discrete trees are exchange rings

Ara¹,

O’Meara²,

Perera³

2003

Trans. Amer. Math. Soc.

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Abstract. It is shown that the Gromov translation ring of a discrete tree over a von Neumann regular ring is an exchange ring. This provides a new source of exchange rings, including, for example, the algebras G(0) of ω × ω matrices (over a field) of constant bandwidth. An extension of these ideas shows that for all real numbers r in the unit interval [0, 1], the growth algebras G(r) (introduced by Hannah and O'Meara in 1993) are exchange rings. Consequently, over a countable field, countable-dimensional exchange algebras can take any prescribed bandwidth dimension r in [0,1].

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“…16E50, 19B10. Very recently, Ara et al [2] showed that the natural homomorphism GL 1 (R) → K 1 (R) is surjective provided that R is a separative exchange ring. A natural problem is the description of the kernel of the epimorphism GL 1 (R) → K 1 (R).…”

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confidence: 99%

Whitehead groups of exchange rings with primitive factors artinian

Chen

2001

International Journal of Mathematics and Mathematical Sciences

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Abstract. We show that if R is an exchange ring with primitive factors artinian then K 1 (R) U(R)/V (R), where U(R) is the group of units of R and V (R) is the subgroup generated by {(1 + ab)(1 + ba) −1 | a, b ∈ R with 1 + ab ∈ U(R)}. As a corollary, K 1 (R) is the abelianized group of units of R if 1/2 ∈ R.2000 Mathematics Subject Classification. 16E50, 19B10. Very recently, Ara et al. [2] showed that the natural homomorphism GL 1 (R) → K 1 (R) is surjective provided that R is a separative exchange ring. A natural problem is the description of the kernel of the epimorphism GL 1 (R) → K 1 (R). In [9, Theorems 1.2 and 1.6], Menal and Moncasi showed that if R satisfies unit 1-stable range or is unitregular, then K 1 (R) U (R)/V (R). Here U(R) is the group of units of R while V (R)is a subgroup described later. In [7], Goodearl and Menal remarked that if for each x, y ∈ R there exists a unit u ∈ R such that x − u and y − u −1 are both units, then K 1 (R) U(R) ab . In this paper, we investigate the above kernel for exchange rings with primitive factors artinian.Recall that R is called an exchange ring if for every right R-module A and two decompositions A = M ⊕ N = ⊕ i∈I A i , where M R R and the index set I is finite, there exist submodules A i ⊆ A i such that A = M ⊕ (⊕ i∈I A i ). It is well known that regular rings, π -regular rings, semiperfect rings, left or right continuous rings, clean rings and unit C * -algebras of real rank zero [1] are all exchange rings.Many authors have studied exchange rings with primitive factors artinian. Fisher and Snider proved that every regular ring with primitive factors artinian is unitregular (see [6, Theorem 6.10]). Moreover, Menal [8, Thereom B] proved that every π -regular ring with primitive factors artinian has stable range one. Recently, Yu [13, Thereom 1] extended these results to exchange rings and showed that every exchange ring with primitive factors artinian has stable range one. On the other hand, Pardo [10] investigated the Grothendieck group K 0 of exchange rings. In this paper, we show that the Whitehead group K 1 (R) U (R)/V (R) for an exchange ring R with primitive factors artinian. We refer the reader to [11] for the general theory of Whitehead groups.Throughout this paper, all rings are associative with identity. The set U(R) denotes the set of all units of R, V (R) denotes the subgroup generated by {p (a, b) (R), the ring of all n × n matrices over R.We start with the following new element-wise property of exchange rings with primitive factors artinian. Lemma 1. Let R be an exchange ring with primitive factors artinian. Then, for any x, y ∈ R, there exists a unit-regular w ∈ R such that 1 + xy − xw ∈ U(R).Proof. Assume that there are some x, y ∈ R such that 1 + xy − xw ∈ U(R) for any unit-regular w ∈ R. Let Ω be the set of all two-sided ideals A of R such that 1 + xy − xw is not a unit modulo A for any unit-regular w + A ∈ R/A. Clearly, Ω ≠ ∅.Given any ascending chain (u+M). As R/M is also exchange, we may assume that e = e 2 ∈ R. Thus we can find posi...

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K₁of separative exchange rings and C*-algebras with real rank zero

Cited by 32 publications

References 28 publications

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

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

Gromov translation algebras over discrete trees are exchange rings

Whitehead groups of exchange rings with primitive factors artinian

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K1of separative exchange rings and C*-algebras with real rank zero

Cited by 32 publications

References 28 publications

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

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

Gromov translation algebras over discrete trees are exchange rings

Whitehead groups of exchange rings with primitive factors artinian

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K₁of separative exchange rings and C*-algebras with real rank zero