Rings of Idempotent Stable Range One

Wang, Zhou; Chen, Jianlong; Khurana, Dinesh; Lam, T. Y.

doi:10.1007/s10468-011-9276-4

Cited by 17 publications

(13 citation statements)

References 12 publications

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“…Unit-endoregular modules are studied recently in [28] which provided some examples and characterizations. In this paper, we provide more characterizations of unit-endoregular modules generalizing results for unit-regular rings (see e.g., Camillo and Khurana [2], Henriksen [12], Wang et al [24]). Although a direct sum of unit-endoregular modules need not be a unit-endoregular module in general, by using the notion of relative endoregularity we provide sufficient and necessary conditions under which the direct sums of unit-endoregular modules are unit-endoregular.…”

Section: Introductionmentioning

confidence: 80%

Characterizations and Direct Sums of Unit-Endoregular Modules

Zhang¹,

Lee

2018

Proceedings of the Edinburgh Mathematical Society

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A module is called unit-endoregular if its endomorphism ring is unit-regular. In this paper, we continue the research in unit-endoregular modules. More characterizations of unit-endoregular modules are obtained. As a special case, we show that for an abelian group G, Endℤ(G) is a unit-regular Baer ring if and only if Endℤ(G) is a two-sided extending regular ring. While the class of unit-endoregular modules is not closed under direct sums, we provide a characterization when there are direct sums of two or more unit-endoregular modules also unit-endoregular under certain conditions. In particular, we investigate unit-endoregular modules which are direct sums of indecomposable modules.

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Section: Introductionmentioning

confidence: 80%

Characterizations and Direct Sums of Unit-Endoregular Modules

Zhang¹,

Lee

2018

Proceedings of the Edinburgh Mathematical Society

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“…(2) Lemma 3.3 provides an easy way to see that regular elements in rings with stable range one are always clean, which was originally proved in [21,Theorem 3.3].…”

Section: Element-wise Connections Between Unit-regularity and Clean Dmentioning

confidence: 99%

Connections between unit-regularity, regularity, cleanness, and strong cleanness of elements and rings

Nielsen

Šter

2017

Trans. Amer. Math. Soc.

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Abstract. We construct an example of a unit-regular ring which is not strongly clean, answering an open question of Nicholson. We also characterize clean matrices with a zero column, and this allows us to describe an interesting connection between unit-regular elements and clean elements. It is also proven that given an element a in a ring R, if a, a 2 , . . . , a k are all regular elements in R (for some k ≥ 1), then there exists w ∈ R such that a i w i a i = a i for 1 ≤ i ≤ k, and a similar statement holds for unit-regular elements. The paper ends with a large number of examples elucidating further connections (and disconnections) between cleanliness, regularity, and unit-regularity.

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“…Recall that a ring R is said to have idempotent stable range one (written isr(R)=1) provided that for any a, b ∈ R, aR + bR = R implies that a + be ∈ U (R) for some e ∈ Id(R) (see [6,16]). If e is an arbitrary element of R (not necessary an idempotent), then R is said to have stable range one.…”

Section: Stable Range Conditionsmentioning

confidence: 99%

“…Clean rings were introduced by Nicholson in relation to exchange rings and have been extensively studied since then. Recently, Wang et al [16] showed that unit regular rings have idempotent stable range one (i.e., whenever aR + bR = R with a, b ∈ R, there exists e 2 = e ∈ R such that a + be ∈ U (R), written isr(R) = 1 for short), and rings with isr(R) = 1 are clean. In 1999, Nicholson [13] called an element of a ring R strongly clean if it is the sum of a unit and an idempotent that commute with each other, and R is strongly clean if each of its elements is strongly clean.…”

Section: Introductionmentioning

confidence: 99%

A Note on Strongly *-Clean Rings

Cui¹,

Wang²

2015

Journal of the Korean Mathematical Society

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Abstract. A * -ring R is called (strongly) * -clean if every element of R is the sum of a projection and a unit (which commute with each other). In this note, some properties of * -clean rings are considered. In particular, a new class of * -clean rings which called strongly π- * -regular are introduced. It is shown that R is strongly π- * -regular if and only if R is π-regular and every idempotent of R is a projection if and only if R/J(R) is strongly regular with J(R) nil, and every idempotent of R/J(R) is lifted to a central projection of R. In addition, the stable range conditions of * -clean rings are discussed, and equivalent conditions among * -rings related to * -cleanness are obtained.

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Rings of Idempotent Stable Range One

Abstract: We show that in a ring of stable range 1, any (von Neumann) regular element is clean. Our main results also imply that any unit-regular ring has idempotent stable range 1 (and is therefore clean), and that a semilocal ring has idempotent stable range 1 if and only if it is semiperfect.

Cited by 17 publications

References 12 publications

Characterizations and Direct Sums of Unit-Endoregular Modules

Characterizations and Direct Sums of Unit-Endoregular Modules

Connections between unit-regularity, regularity, cleanness, and strong cleanness of elements and rings

A Note on Strongly *-Clean Rings

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