Stable range one for rings with many idempotents

Camillo, Victor; Yu, Hua-Ping

doi:10.1090/s0002-9947-1995-1277100-2

Cited by 42 publications

(16 citation statements)

References 13 publications

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“…Then by [7,Lemma 6] there is a positive integer n and an element x ∈ vL K (E)v such that ax = xa and a n+1 x = a n = xa n+1 . Now iterating the substitution a n = a n+1 x = aa n x = a(a n+1 x)x = a n+2 x 2 we get a n = a 2n x n , which using ax = xa gives a n = a n x n a n , which yields (2).…”

Section: Then It Is Clear By Construction That B(s 1 ) ∪ B(s 2 ) ⊆ B(mentioning

confidence: 99%

Regularity Conditions for Arbitrary Leavitt Path Algebras

Abrams

Rangaswamy

2009

Algebr Represent Theor

115

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We show that if E is an arbitrary acyclic graph then the Leavitt path algebra L K (E) is locally K-matricial; that is, L K (E) is the direct union of subalgebras, each isomorphic to a finite direct sum of finite matrix rings over the field K. (Here an arbitrary graph means that neither cardinality conditions nor graph-theoretic conditions (e.g. row-finiteness) are imposed on E. These unrestrictive conditions are in contrast to the hypotheses used in much of the literature on this subject.) As a consequence we get our main result, in which we show that the following conditions are equivalent for an arbitrary graph E:is strongly π-regular. We conclude by showing how additional regularity conditions (unit regularity, strongly clean) can be appended to this list of equivalent conditions.

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Section: Then It Is Clear By Construction That B(s 1 ) ∪ B(s 2 ) ⊆ B(mentioning

confidence: 99%

Regularity Conditions for Arbitrary Leavitt Path Algebras

Abrams

Rangaswamy

2009

Algebr Represent Theor

115

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“…More recently, Yu [20], [21] and Camillo and Yu [5] proved that strongly π-regular rings have stable range one under some additional hypothesis. For example they show that a strongly π-regular ring such that every power of a regular element is regular, has stable range one [5,Theorem 5], generalizing [11,Theorem 5.8].…”

Section: Introductionmentioning

confidence: 99%

“…For example they show that a strongly π-regular ring such that every power of a regular element is regular, has stable range one [5,Theorem 5], generalizing [11,Theorem 5.8].…”

Section: Introductionmentioning

confidence: 99%

Strongly 𝜋-regular rings have stable range one

Ara¹

1996

Proc. Amer. Math. Soc.

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Abstract. A ring R is said to be strongly π-regular if for every a ∈ R there exist a positive integer n and b ∈ R such that a n = a n+1 b. For example, all algebraic algebras over a field are strongly π-regular. We prove that every strongly π-regular ring has stable range one. The stable range one condition is especially interesting because of Evans' Theorem, which states that a module M cancels from direct sums whenever End R (M ) has stable range one. As a consequence of our main result and Evans' Theorem, modules satisfying Fitting's Lemma cancel from direct sums.

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“…[8,13]). Many authors have studied stable range one conditions over exchange rings such as [1,2,4,5,6,7,15,16].In this paper, we investigate stable range one conditions over exchange rings by virtue of Drazin inverses, nilpotent elements, and prime ideals. We showed that stable range conditions can be determined by Drazin inverses for exchange rings.…”

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confidence: 99%

Exchange rings having stable range one

Chen

2001

International Journal of Mathematics and Mathematical Sciences

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Abstract. We investigate the sufficient conditions and the necessary conditions on an exchange ring R under which R has stable range one. These give nontrivial generalizations of Theorem 3 of V. P. Camillo and H.-P. Yu (1995), Theorem 4.19 of K. R. Goodearl (1979Goodearl ( , 1991, Theorem 2 of R. E. Hartwig (1982), and Theorem 9 of H.-P. Yu (1995). We call R has stable range one provided that aR+bR = R implies that a+by ∈ U(R) for a y ∈ R. It is well known that an exchange ring R has stable range one if and only if A ⊕ B A ⊕ C implies B C for all finitely generated projective right R-modules A, B, and C. It has been realized that the class of rings having stable range one has good stability properties in a K-theoretic sense (cf. [8,13]). Many authors have studied stable range one conditions over exchange rings such as [1,2,4,5,6,7,15,16].In this paper, we investigate stable range one conditions over exchange rings by virtue of Drazin inverses, nilpotent elements, and prime ideals. We showed that stable range conditions can be determined by Drazin inverses for exchange rings. Also, we see that these stable range conditions can be determined by regular elements out of any proper ideal of R. Moreover, we prove that an exchange ring R has stable range one if the set of nilpotents is closed under product. These extend the corresponding results of [4, Theorem 3], [9, Theorem 4.19], [12, Theorem 2],and [16, Theorem 9].Throughout this paper, all rings are associative ring with identities and all right R-modules are unitary right R-modules. M ⊕ N means that right R-module M is isomorphic to a direct summand of right R-module N. The notation x ≈ y means that x = uyu −1 for some u ∈ U(R), where U(R) denotes the set of all units of R. Call a ∈ R is regular if a = axa for some x ∈ R and a ∈ R is unit-regular if a = aua for some u ∈ U(R).

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Stable range one for rings with many idempotents

Cited by 42 publications

References 13 publications

Regularity Conditions for Arbitrary Leavitt Path Algebras

Regularity Conditions for Arbitrary Leavitt Path Algebras

Strongly 𝜋-regular rings have stable range one

Exchange rings having stable range one

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