A Class of Quasipolar Rings

Cui, Jian; Chen, Jianlong

doi:10.1080/00927872.2011.610854

Cited by 14 publications

(14 citation statements)

References 13 publications

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“…Furthermore, we show that strong J-cleanness for triangular matrix rings over a local ring can be enhanced to such stronger properties. These also generalize the corresponding properties of J-quasipolarity, e.g., [8,Theorem 4.9].…”

Section: Introductionmentioning

confidence: 61%

“…Therefore the proof is complete by Corollary 4.2. ✷ Following Cui and Chen [8], a ring R is called J-quasipolar provided that for any element a ∈ R there exists some e ∈ comm 2 (a) such that a + e ∈ J(R). We further show that the two concepts coincide with each other.…”

Section: Corollary 47 a Ring R Is Perfectly J-clean If And Only Ifmentioning

confidence: 99%

“…Recently, quasipolar 2 × 2 matrix rings and triangular matrix rings over local rings are also studied from different point of views (cf. [7][8][9] and [11]). The motivation of this article is to introduce a medium class between strongly clean rings and quasipolar rings.…”

Section: Introductionmentioning

confidence: 99%

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On Perfectly Clean Rings

Chen¹,

Halicioglu²,

Kose³

2013

Preprint

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An element a of a ring R is called perfectly clean if there exists an idempotent e ∈ comm 2 (a) such that a − e ∈ U (R). A ring R is perfectly clean in case every element in R is perfectly clean. In this paper, we investigate conditions on a local ring R that imply that 2 × 2 matrix rings and triangular matrix rings are perfectly clean. We shall show that for these rings perfect cleanness and strong cleanness coincide with each other, and enhance many known results. We also obtain several criteria for such a triangular matrix ring to be perfectly J-clean. For instance, it is proved that for a commutative ring R, Tn(R) is perfectly J-clean if and only if R is strongly J-clean.

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Section: Introductionmentioning

confidence: 61%

Section: Corollary 47 a Ring R Is Perfectly J-clean If And Only Ifmentioning

confidence: 99%

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On Perfectly Clean Rings

Chen¹,

Halicioglu²,

Kose³

2013

Preprint

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“…Conversely, if u 1 au is ı-quasipolar, then by the preceding proof In [4,Corollary 2.3], it is proved that if R is a J -quasipolar ring, then 2 2 J.R/. In this direction we prove the following.…”

Section: ı-Quasipolar Ringsmentioning

confidence: 78%

“…The ring R is said to be nil-quasipolar if every element of R is nilquasipolar. Recently, J -quasipolar rings are studied in [4]. The element a is called J -quasipolar if there exists p 2 D p 2 R such that p 2 comm 2 .a/ and a C p 2 J.R/, p is called a J -spectral idempotent of a.…”

Section: Introductionmentioning

confidence: 99%

A Generalization of $J$-Quasipolar Rings

Calci¹,

Halicioğlu²,

Harmanci³

2017

MMN

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Abstract. In this paper we introduce a class of quasipolar rings which is a generalization of Jquasipolar rings. Let R be a ring with identity. An element a 2 R is called ı-quasipolar if there exists p 2 D p 2 comm 2 .a/ such that a C p is contained in ı.R/, and the ring R is called ı-quasipolar if every element of R is ı-quasipolar. We use ı-quasipolar rings to extend some results of J -quasipolar rings. Then some of the main results of J -quasipolar rings are special cases of our results for this general setting. We give many characterizations and investigate general properties of ı-quasipolar rings.

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