Strong cleanness of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mn>2</mml:mn><mml:mo>×</mml:mo><mml:mn>2</mml:mn></mml:math> matrix ring over a general local ring

Yang, Xiande; Zhou, Yiqiang

doi:10.1016/j.jalgebra.2008.06.012

Journal of Algebra

2008

DOI: 10.1016/j.jalgebra.2008.06.012

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Strong cleanness of the 2×2 matrix ring over a general local ring

Xiande Yang

Yiqiang Zhou

Abstract: A ring R is called strongly clean if every element of R is the sum of a unit and an idempotent that commute with each other. A recent result of Borooah, Diesl and Dorsey [G. Borooah, A.J. Diesl, T.J. Dorsey, Strongly clean matrix rings over commutative local rings, J. Pure Appl. Algebra 212 (1) (2008) 281-296] completely characterized the commutative local rings R for which M n (R) is strongly clean. For a general local ring R and n > 1, however, it is unknown when the matrix ring M n (R) is strongly clean. He… Show more

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Cited by 29 publications

References 11 publications

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A Survey of Strongly Clean Rings

Yang

2008

Acta Appl Math

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Let R be an associative ring with identity. An element a ∈ R is called clean if a = e + u with e an idempotent and u a unit of R and a is called strongly clean if, in addition, eu = ue. A ring R is called clean if every element of R is clean and R is strongly clean if every element of R is strongly clean. In the paper [Nicholson and Zhou, Clean rings: a survey, Advances in Ring Theory, 181-198, World Sci. Pub., Hackensack, NJ, 2005], the authors brought out an up to date account of the results in the study of clean rings. Here, we give an account of the results on strongly clean rings.

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A Survey of Strongly Clean Rings

Yang

2008

Acta Appl Math

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sp-Groups and Their Endomorphism Rings

2021

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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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Strong cleanness of the 2×2 matrix ring over a general local ring

Cited by 29 publications

References 11 publications

A Survey of Strongly Clean Rings

A Survey of Strongly Clean Rings

sp-Groups and Their Endomorphism Rings

Nil-quasipolar rings

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