On Strongly<i>J</i>-Clean Rings

Chen, Huanyin

doi:10.1080/00927870903286835

Cited by 35 publications

(31 citation statements)

References 10 publications

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“…According to Lemma 3.5, C is nil-quasipolar, and so is A by Lemma 3.1. [3,Proposition 5.8] we have the following results for nil-quasipolar rings. (1) Every purely singular matrix in M 2 (R) is nil-quasipolar.…”

Section: ) χ(A) = T 2 −Tr(a)t +Det (A) = 0 Has a Root In N Il(r) Andmentioning

confidence: 96%

Nil-quasipolar rings

Gurgun

Halicioğlu

Harmanci

2014

Bol. Soc. Mat. Mex.

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Let R be an arbitrary ring. An element a ∈ R is nil-quasipolar if there exists p 2 = p ∈ comm 2 (a) such that a + p ∈ Nil(R); R is called nil-quasipolar in case each of its elements is nil-quasipolar. In this paper, we study nil-quasipolar rings over commutative local rings. We determine the conditions under which a single 2 × 2 matrix over a commutative local ring is nil-quasipolar. It is shown that A ∈ M 2 (R) is nil-quasipolar if and only if A ∈ Nil M 2 (R) or A + I 2 ∈ Nil M 2 (R) or the characteristic polynomial χ(A) has a root in Nil(R) and a root in −1+Nil(R). We give some equivalent characterizations of nil-quasipolar rings through the endomorphism ring of a module. Among others we prove that every nil-quasipolar ring has stable range one.

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Section: ) χ(A) = T 2 −Tr(a)t +Det (A) = 0 Has a Root In N Il(r) Andmentioning

confidence: 96%

Nil-quasipolar rings

Gurgun

Halicioğlu

Harmanci

2014

Bol. Soc. Mat. Mex.

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“…Strongly J -clean rings were introduced by Chen in [3]. For a ring R the element a 2 R is called J -clean if a is the sum of an idempotent and a radical element in its Jacobson radical.…”

Section: ı-Quasipolar Ringsmentioning

confidence: 99%

“…An element of a ring is called strongly J -clean [3] provided that it can be written as the sum of an idempotent and an element in its Jacobson radical that commute. A ring is strongly J -clean in case each of its elements is strongly J -clean.…”

Section: Weakly ı-Quasipolar Ringsmentioning

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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“…Chen [6] calls a ring R strongly J -clean if for every element a ∈ R there exists an idempotent e ∈ R such that a − e ∈ J and ea = ae. Call a ring R J -clean if for any element a ∈ R , there exists an idempotent e ∈ R such that a − e ∈ J .…”

Section: δ R -Clean Ringsmentioning

confidence: 99%

A class of uniquely (strongly) clean rings

Gurgun¹,

Özcan²

2014

Turk J Math

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In this paper we call a ring R δr -clean if every element is the sum of an idempotent and an element in δ(RR)where δ(RR) is the intersection of all essential maximal right ideals of R . If this representation is unique (and the elements commute) for every element we call the ring uniquely (strongly) δr -clean. Various basic characterizations and properties of these rings are proved, and many extensions are investigated and many examples are given. In particular, we see that the class of δr -clean rings lies between the class of uniquely clean rings and the class of exchange rings, and the class of uniquely strongly δr -clean rings is a subclass of the class of uniquely strongly clean rings. We prove that R is δr -clean if and only if R/δr(RR) is Boolean and R/Soc(RR) is clean where Soc(RR) is the right socle of R .

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On StronglyJ-Clean Rings

Cited by 35 publications

References 10 publications

Nil-quasipolar rings

Nil-quasipolar rings

A Generalization of $J$-Quasipolar Rings

A class of uniquely (strongly) clean rings

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