Stable range one for rings with many units

Goodearl, K. R.; Menal, Pere

doi:10.1016/0022-4049(88)90034-5

Cited by 92 publications

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“…Theorem 5: If R is a strongly w-regular ring with the property that all powers of every regular element are regular, then R has stable range 1. The latter generalizes a recent result of Goodearl and Menai [5].Let R be an associative ring with identity. R is said to have stable range 1 if for any a, b £ R satisfying aR + bR = R, there exists y £ R such that a + by is a unit.…”

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“…It is an open question whether all strongly 7r-regular rings have stable range 1. Goodearl and Menai [5] proved that strongly ^-regular rings are unit-regular and, hence, have stable range 1 (Theorem 5.8, p. 278).In this note we first extend the above result for von Neumann regular rings to a larger class of rings, which includes all strongly 7r-regular rings, 7t-regular rings, von Neumann regular rings, and algebraic algebras. As an application of this, we prove that a strongly ^-regular ring P has stable range 1 if powers of every regular element are regular.…”

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Stable Range One for Rings with many Idempotents

Camillo¹,

Yu²

1995

Transactions of the American Mathematical Society

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Abstract.An associative ring R is said to have stable range 1 if for any a, b e R satisfying aR + bR = R , there exists y e R such that a + by is a unit. The purpose of this note is to prove the following facts. Theorem 3: An exchange ring R has stable range 1 if and only if every regular element of R is unit-regular. Theorem 5: If R is a strongly w-regular ring with the property that all powers of every regular element are regular, then R has stable range 1. The latter generalizes a recent result of Goodearl and Menai [5].Let R be an associative ring with identity. R is said to have stable range 1 if for any a, b £ R satisfying aR + bR = R, there exists y £ R such that a + by is a unit. This definition is left-right symmetric by Vaserstein [9, Theorem 2]. Furthermore, by a theorem of Kaplansky, all one-sided units are two-sided in rings having stable range 1 (cf. Vaserstein [10, Theorem 2.6]). It is well known that a (von Neumann) regular ring P has stable range 1 if and only if R is unit-regular (see, for example, Goodearl [4, Proposition 4.12]).Call a ring P strongly n-regular if for every element a £ R there exist a number n (depending on a) and an element x £ R such that a" -an+lx. This is in fact a two-sided condition [3]. It is an open question whether all strongly 7r-regular rings have stable range 1. Goodearl and Menai [5] proved that strongly ^-regular rings are unit-regular and, hence, have stable range 1 (Theorem 5.8, p. 278).In this note we first extend the above result for von Neumann regular rings to a larger class of rings, which includes all strongly 7r-regular rings, 7t-regular rings, von Neumann regular rings, and algebraic algebras. As an application of this, we prove that a strongly ^-regular ring P has stable range 1 if powers of every regular element are regular. The latter is a generalization of the abovementioned result of Goodearl and Menai for strongly 7r-regular regular rings. As one can see from our proofs, rings in these classes have a large supply of idempotents.Throughout, R stands for an associative ring with identity and J(R) for the Jacobson radical of R . Modules are unitary right P-modules except otherwise specified. For other undefined terms, readers are referred to [4].Let Mr be a right P-module. Following Crawley and Jonsson [2], Mr is said to have the exchange property if for every module Ar and any two

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Stable Range One for Rings with many Idempotents

Camillo¹,

Yu²

1995

Transactions of the American Mathematical Society

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“…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.…”

Section: (R) U (R)/v (R) Here U(r) Is the Group Of Units Of R While mentioning

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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Von Neumann Regular Rings and Direct Sum Decomposition Problems

Goodearl

1995

Abelian Groups and Modules

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Stable range one for rings with many units

Cited by 92 publications

References 15 publications

Stable Range One for Rings with many Idempotents

Stable Range One for Rings with many Idempotents

Whitehead groups of exchange rings with primitive factors artinian

Von Neumann Regular Rings and Direct Sum Decomposition Problems

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