Some families of strongly clean rings

Yang, Xiande; Zhou, Yiqiang

doi:10.1016/j.laa.2007.03.012

Cited by 17 publications

(6 citation statements)

References 12 publications

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“…Several authors discussed when the matrix ring M n (R) over a commutative local ring R is strongly clean (see [10,17,18,22,28,32,43,44]). …”

Section: Strongly Clean Matrix Rings Over Commutative Ringsmentioning

confidence: 99%

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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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“…Several authors discussed when the matrix ring M n (R) over a commutative local ring R is strongly clean (see [10,17,18,22,28,32,43,44]). …”

Section: Strongly Clean Matrix Rings Over Commutative Ringsmentioning

confidence: 99%

“…In [28,44], the authors discussed if the strongly clean property of M n (R) is equivalent to that of M n (R [[x]]) and that of M n ( R [x] (x k ) ). Theorem 5.22 can be used to prove the next result.…”

Section: Definition 519mentioning

confidence: 99%

A Survey of Strongly Clean Rings

Yang

2008

Acta Appl Math

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“…Strongly clean matrices over commutative local rings were extensively studied by many authors from very different view points (cf. [1][2], [4][5][6][7] and [9][10][11][12]). In [5], Diesl introduced the concept of strongly nil cleanness.…”

Section: Introductionmentioning

confidence: 99%

STRONGLY NIL CLEAN MATRICES OVER R[x]/(x²-1)

Chen¹

2012

Bulletin of the Korean Mathematical Society

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“…Strongly clean generalized 2 × 2 matrices over a local ring were also studied by Tang and Zhou (cf. [10,Theorem 15]. In [1,Theorem 12], Borooah et al characterized when an n × n matrix over a commutative local ring R is strongly clean, in terms of factorization in the polynomial ring R [t].…”

Section: Introductionmentioning

confidence: 99%

Strongly Clean Matrices Over Commutative Local Rings

2012

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A commutative ring R is projective free provided that every finitely generated Rmodule is free. An element in a ring is strongly clean provided that it is the sum of an idempotent and a unit that commutates. Let R be a projective-free ring, and let h ∈ R[t] be a monic polynomial of degree n. We prove, in this article, that every ϕ ∈ Mn(R) with characteristic polynomial h is strongly clean, if and only if the companion matrix C h of h is strongly clean, if and only if there exists a factorization h = h0h1 such that h0 ∈ S0, h1 ∈ S1 and (h0, h1) = 1. Matrices over power series over projective rings are also discussed. These extend the known results [1, Theorem 12] and [5, Theorem 25].

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Some families of strongly clean rings

Cited by 17 publications

References 12 publications

A Survey of Strongly Clean Rings

A Survey of Strongly Clean Rings

STRONGLY NIL CLEAN MATRICES OVER R[x]/(x²-1)

Strongly Clean Matrices Over Commutative Local Rings

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Some families of strongly clean rings

Cited by 17 publications

References 12 publications

A Survey of Strongly Clean Rings

A Survey of Strongly Clean Rings

STRONGLY NIL CLEAN MATRICES OVER R[x]/(x2-1)

Strongly Clean Matrices Over Commutative Local Rings

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Product

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STRONGLY NIL CLEAN MATRICES OVER R[x]/(x²-1)