• 제목/요약/키워드: Strongly clean matrix

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STRONGLY CLEAN MATRIX RINGS OVER NONCOMMUTATIVE LOCAL RINGS

  • Li, Bingjun
    • 대한수학회보
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    • 제46권1호
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    • pp.71-78
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    • 2009
  • An element of a ring R with identity is called strongly clean if it is the sum of an idempotent and a unit that commute, and R is called strongly clean if every element of R is strongly clean. Let R be a noncommutative local ring, a criterion in terms of solvability of a simple quadratic equation in R is obtained for $M_2$(R) to be strongly clean.

Certain Clean Decompositions for Matrices over Local Rings

  • Yosum Kurtulmaz;Handan Kose;Huanyin Chen
    • Kyungpook Mathematical Journal
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    • 제63권4호
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    • pp.561-569
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    • 2023
  • An element a ∈ R is strongly rad-clean provided that there exists an idempotent e ∈ R such that a - e ∈ U(R), ae = ea and eae ∈ J(eRe). In this article, we completely determine when a 2 × 2 matrix over a commutative local ring is strongly rad clean. An application to matrices over power-series is also given.

Strongly Clean Matrices Over Power Series

  • Chen, Huanyin;Kose, Handan;Kurtulmaz, Yosum
    • Kyungpook Mathematical Journal
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    • 제56권2호
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    • pp.387-396
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    • 2016
  • An $n{\times}n$ matrix A over a commutative ring is strongly clean provided that it can be written as the sum of an idempotent matrix and an invertible matrix that commute. Let R be an arbitrary commutative ring, and let $A(x){\in}M_n(R[[x]])$. We prove, in this note, that $A(x){\in}M_n(R[[x]])$ is strongly clean if and only if $A(0){\in}M_n(R)$ is strongly clean. Strongly clean matrices over quotient rings of power series are also determined.

SOME STRONGLY NIL CLEAN MATRICES OVER LOCAL RINGS

  • Chen, Huanyin
    • 대한수학회보
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    • 제48권4호
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    • pp.759-767
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    • 2011
  • An element of a ring is called strongly nil clean provided that it can be written as the sum of an idempotent and a nilpotent element that commute. A ring is strongly nil clean in case each of its elements is strongly nil clean. We investigate, in this article, the strongly nil cleanness of 2${\times}$2 matrices over local rings. For commutative local rings, we characterize strongly nil cleanness in terms of solvability of quadratic equations. The strongly nil cleanness of a single triangular matrix is studied as well.

ON 2 × 2 STRONGLY CLEAN MATRICES

  • Chen, Huanyin
    • 대한수학회보
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    • 제50권1호
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    • pp.125-134
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    • 2013
  • An element in a ring R is strongly clean provided that it is the sum of an idempotent and a unit that commutate. In this note, several necessary and sufficient conditions under which a $2{\times}2$ matrix over an integral domain is strongly clean are given. These show that strong cleanness over integral domains can be characterized by quadratic and Diophantine equations.

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

  • Chen, Huanyin
    • 대한수학회보
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    • 제49권3호
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    • pp.589-599
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    • 2012
  • An element of a ring is called strongly nil clean provided that it can be written as the sum of an idempotent and a nilpotent element that commute. We characterize, in this article, the strongly nil cleanness of $2{\times}2$ and $3{\times}3$ matrices over $R[x]/(x^2-1)$ where $R$ is a commutative local ring with characteristic 2. Matrix decompositions over fields are derived as special cases.

STRONG P-CLEANNESS OF TRIVIAL MORITA CONTEXTS

  • Calci, Mete B.;Halicioglu, Sait;Harmanci, Abdullah
    • 대한수학회논문집
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    • 제34권4호
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    • pp.1069-1078
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    • 2019
  • Let R be a ring with identity and P(R) denote the prime radical of R. An element r of a ring R is called strongly P-clean, if there exists an idempotent e such that $r-e=p{\in}P$(R) with ep = pe. In this paper, we determine necessary and sufficient conditions for an element of a trivial Morita context to be strongly P-clean.

SUMS OF TRIPOTENT AND NILPOTENT MATRICES

  • Abdolyousefi, Marjan Sheibani;Chen, Huanyin
    • 대한수학회보
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    • 제55권3호
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    • pp.913-920
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    • 2018
  • Let R be a 2-primal strongly 2-nil-clean ring. We prove that every square matrix over R is the sum of a tripotent and a nilpotent matrices. The similar result for rings of bounded index is proved. We thereby provide a large class of rings over which every matrix is the sum of a tripotent and a nilpotent matrices.

QUASIPOLAR MATRIX RINGS OVER LOCAL RINGS

  • Cui, Jian;Yin, Xiaobin
    • 대한수학회보
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    • 제51권3호
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    • pp.813-822
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    • 2014
  • A ring R is called quasipolar if for every a 2 R there exists $p^2=p{\in}R$ such that $p{\in}comm^2{_R}(a)$, $ a+p{\in}U(R)$ and $ap{\in}R^{qnil}$. The class of quasipolar rings lies properly between the class of strongly ${\pi}$-regular rings and the class of strongly clean rings. In this paper, we determine when a $2{\times}2$ matrix over a local ring is quasipolar. Necessary and sufficient conditions for a $2{\times}2$ matrix ring to be quasipolar are obtained.