• Title, Summary, Keyword: matrix ring

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ON NILPOTENT-DUO RINGS

  • Piao, Zhelin
    • Journal of the Chungcheong Mathematical Society
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    • v.32 no.4
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    • pp.401-408
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    • 2019
  • A ring R is called right (resp., left) nilpotent-duo if N(R)a ⊆ aN(R) (resp., aN(R) ⊆ N(R)a) for every a ∈ R, where N(R) is the set of all nilpotents in R. In this article we provide other proofs of known results and other computations for known examples in relation with right nilpotent-duo property. Furthermore we show that the left nilpotent-duo property does not go up to a kind of matrix ring.

I-RINGS AND TRIANGULAR MATRIX RINGS

  • Min, Kang-Joo
    • Journal of the Chungcheong Mathematical Society
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    • v.14 no.2
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    • pp.19-26
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    • 2001
  • All rings are assumed to be associative but do not necessarily have an identity. In this paper, we carry out a study of ring theoretic properties of formal triangular matrix rings. Some results are obtained on these rings concerning properties such as being $I_0$-ring, I-ring, exchange ring.

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QUASIPOLAR MATRIX RINGS OVER LOCAL RINGS

  • Cui, Jian;Yin, Xiaobin
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.3
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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.

GENERALIZED CAYLEY GRAPH OF UPPER TRIANGULAR MATRIX RINGS

  • Afkhami, Mojgan;Hashemifar, Seyed Hosein;Khashyarmanesh, Kazem
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.4
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    • pp.1017-1031
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    • 2016
  • Let R be a commutative ring with the non-zero identity and n be a natural number. ${\Gamma}^n_R$ is a simple graph with $R^n{\setminus}\{0\}$ as the vertex set and two distinct vertices X and Y in $R^n$ are adjacent if and only if there exists an $n{\times}n$ lower triangular matrix A over R whose entries on the main diagonal are non-zero such that $AX^t=Y^t$ or $AY^t=X^t$, where, for a matrix B, $B^t$ is the matrix transpose of B. ${\Gamma}^n_R$ is a generalization of Cayley graph. Let $T_n(R)$ denote the $n{\times}n$ upper triangular matrix ring over R. In this paper, for an arbitrary ring R, we investigate the properties of the graph ${\Gamma}^n_{T_n(R)}$.

ON A GENERALIZATION OF THE MCCOY CONDITION

  • Jeon, Young-Cheol;Kim, Hong-Kee;Kim, Nam-Kyun;Kwak, Tai-Keun;Lee, Yang;Yeo, Dong-Eun
    • Journal of the Korean Mathematical Society
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    • v.47 no.6
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    • pp.1269-1282
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    • 2010
  • We in this note consider a new concept, so called $\pi$-McCoy, which unifies McCoy rings and IFP rings. The classes of McCoy rings and IFP rings do not contain full matrix rings and upper (lower) triangular matrix rings, but the class of $\pi$-McCoy rings contain upper (lower) triangular matrix rings and many kinds of full matrix rings. We first study the basic structure of $\pi$-McCoy rings, observing the relations among $\pi$-McCoy rings, Abelian rings, 2-primal rings, directly finite rings, and ($\pi-$)regular rings. It is proved that the n by n full matrix rings ($n\geq2$) over reduced rings are not $\pi$-McCoy, finding $\pi$-McCoy matrix rings over non-reduced rings. It is shown that the $\pi$-McCoyness is preserved by polynomial rings (when they are of bounded index of nilpotency) and classical quotient rings. Several kinds of extensions of $\pi$-McCoy rings are also examined.

ON THE FI-EXTENDING MODULES

  • Min, Kang-Joo
    • Journal of the Chungcheong Mathematical Society
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    • v.16 no.2
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    • pp.79-88
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    • 2003
  • In this paper, we study properties of a free normalizing extension ring of a FI-extending ring. We develop properties of formal triangular matrix rings and FI-extending rings. Several results on the quasi-extending modules are obtained.

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SUMS OF TRIPOTENT AND NILPOTENT MATRICES

  • Abdolyousefi, Marjan Sheibani;Chen, Huanyin
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.3
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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.

ELEMENTARY MATRIX REDUCTION OVER ZABAVSKY RINGS

  • Chen, Huanyin;Sheibani, Marjan
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.1
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    • pp.195-204
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    • 2016
  • We prove, in this note, that a Zabavsky ring R is an elementary divisor ring if and only if R is a $B{\acute{e}}zout$ ring. Many known results are thereby generalized to much wider class of rings, e.g. [4, Theorem 14], [7, Theorem 4], [9, Theorem 1.2.14], [11, Theorem 4] and [12, Theorem 7].

A QUESTION ON ⁎-REGULAR RINGS

  • Cui, Jian;Yin, Xiaobin
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.5
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    • pp.1333-1338
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    • 2018
  • A ${\ast}-ring$ R is called ${\ast}-regular$ if every principal one-sided ideal of R is generated by a projection. In this note, several characterizations of ${\ast}-regular$ rings are provided. In particular, it is shown that a matrix ring $M_n(R)$ is ${\ast}-regular$ if and only if R is regular and $1+x^*_1x_1+{\cdots}+x^*_{n-1}x_{n-1}$ is a unit for all $x_i$ of R; which answers a question raised in the literature recently.

STRONGLY CLEAN MATRIX RINGS OVER NONCOMMUTATIVE LOCAL RINGS

  • Li, Bingjun
    • Bulletin of the Korean Mathematical Society
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    • v.46 no.1
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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.