• Title, Summary, Keyword: von Neumann regular rings

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A Note on GQ-injectivity

  • Kim, Jin-Yong
    • Kyungpook Mathematical Journal
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    • v.49 no.2
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    • pp.389-392
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    • 2009
  • The purpose of this note is to improve several known results on GQ-injective rings. We investigate in this paper the von Neumann regularity of left GQ-injective rings. We give an answer a question of Ming in the positive. Actually it is proved that if R is a left GQ-injective ring whose simple singular left R-modules are GP-injective then R is a von Neumann regular ring.

ON φ-VON NEUMANN REGULAR RINGS

  • Zhao, Wei;Wang, Fanggui;Tang, Gaohua
    • Journal of the Korean Mathematical Society
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    • v.50 no.1
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    • pp.219-229
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    • 2013
  • Let R be a commutative ring with $1{\neq}0$ and let $\mathcal{H}$ = {R|R is a commutative ring and Nil(R) is a divided prime ideal}. If $R{\in}\mathcal{H}$, then R is called a ${\phi}$-ring. In this paper, we introduce the concepts of ${\phi}$-torsion modules, ${\phi}$-flat modules, and ${\phi}$-von Neumann regular rings.

On SF-rings and Regular Rings

  • Subedi, Tikaram;Buhphang, Ardeline Mary
    • Kyungpook Mathematical Journal
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    • v.53 no.3
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    • pp.397-406
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    • 2013
  • A ring R is called a left (right) SF-ring if simple left (right) R-modules are flat. It is still unknown whether a left (right) SF-ring is von Neumann regular. In this paper, we give some conditions for a left (right) SF-ring to be (a) von Neumann regular; (b) strongly regular; (c) division ring. It is proved that: (1) a right SF-ring R is regular if maximal essential right (left) ideals of R are weakly left (right) ideals of R (this result gives an affirmative answer to the question raised by Zhang in 1994); (2) a left SF-ring R is strongly regular if every non-zero left (right) ideal of R contains a non-zero left (right) ideal of R which is a W-ideal; (3) if R is a left SF-ring such that $l(x)(r(x))$ is an essential left (right) ideal for every right (left) zero divisor x of R, then R is a division ring.

ON NCI RINGS

  • Hwang, Seo-Un;Jeon, Young-Cheol;Park, Kwang-Sug
    • Bulletin of the Korean Mathematical Society
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    • v.44 no.2
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    • pp.215-223
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    • 2007
  • We in this note introduce the concept of NCI rings which is a generalization of NI rings. We study the basic structure of NCI rings, concentrating rings of bounded index of nilpotency and von Neumann regular rings. We also construct suitable examples to the situations raised naturally in the process.

WEAKLY (m, n)-CLOSED IDEALS AND (m, n)-VON NEUMANN REGULAR RINGS

  • Anderson, David F.;Badawi, Ayman;Fahid, Brahim
    • Journal of the Korean Mathematical Society
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    • v.55 no.5
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    • pp.1031-1043
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    • 2018
  • Let R be a commutative ring with $1{\neq}0$, I a proper ideal of R, and m and n positive integers. In this paper, we define I to be a weakly (m, n)-closed ideal if $0{\neq}x^m\;{\in}I$ for $x{\in}R$ implies $x^n{\in}I$, and R to be an (m, n)-von Neumann regular ring if for every $x{\in}R$, there is an $r{\in}R$ such that $x^mr=x^n$. A number of results concerning weakly(m, n)-closed ideals and (m, n)-von Neumann regular rings are given.

ON RINGS CONTAINING A P-INJECTIVE MAXIMAL LEFT IDEAL

  • Kim, Jin-Yong;Kim, Nam-Kyun
    • Communications of the Korean Mathematical Society
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    • v.18 no.4
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    • pp.629-633
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    • 2003
  • We investigate in this paper rings containing a finitely generated p-injective maximal left ideal. We show that if R is a semiprime ring containing a finitely generated p-injective maximal left ideal, then R is a left p-injective ring. Using this result we are able to give a new characterization of von Neumann regular rings with nonzero socle.

On Left SF-Rings and Strongly Regular Rings

  • Subedi, Tikaram;Buhphang, Ardeline Mary
    • Kyungpook Mathematical Journal
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    • v.56 no.3
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    • pp.861-866
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    • 2016
  • A ring R called left SF if its simple left modules are at. Regular rings are known to be left SF-rings. However, till date it is unknown whether a left SF-ring is necessarily regular. In this paper, we prove the strong regularity of left (right) complement bounded left SF-rings. We also prove the strong regularity of a class of generalized semi-commutative left SF-rings.

A NOTE ON SIMPLE SINGULAR GP-INJECTIVE MODULES

  • Nam, Sang Bok
    • Korean Journal of Mathematics
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    • v.7 no.2
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    • pp.215-218
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    • 1999
  • We investigate characterizations of rings whose simple singular right R-modules are GP-injective. It is proved that if R is a semiprime ring whose simple singular right R-modules are GP-injective, then the center $Z(R)$ of R is a von Neumann regular ring. We consider the condition ($^*$): R satisfies $l(a){\subseteq}r(a)$ for any $a{\in}R$. Also it is shown that if R satisfies ($^*$) and every simple singular right R-module is GP-injective, then R is a reduced weakly regular ring.

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