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Rings having normality in terms of the Jacobson radical

  • H. Kose
  • Y. Kurtulmaz
  • B. Ungor
  • A. HarmanciEmail author
Open Access
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Abstract

A ring R is defined to be J-normal if for any \(a, r\in R\) and idempotent \(e\in R\), \(ae = 0\) implies \(Rera\subseteq J(R)\), where J(R) is the Jacobson radical of R. The class of J-normal rings lies between the classes of weakly normal rings and left min-abel rings. It is proved that R is J-normal if and only if for any idempotent \(e\in R\) and for any \(r\in R\), \(R(1 - e)re\subseteq J(R)\) if and only if for any \(n\ge 1\), the \(n\times n\) upper triangular matrix ring \(U_{n}(R)\) is a J-normal ring if and only if the Dorroh extension of R by \({\mathbb {Z}}\) is J-normal. We show that R is strongly regular if and only if R is J-normal and von Neumann regular. For a J-normal ring R, it is obtained that R is clean if and only if R is exchange. We also investigate J-normality of certain subrings of the ring of \(2\times 2\) matrices over R.

Mathematics Subject Classification

16D25 16N20 16U99 

Notes

Acknowledgements

The authors would like to thank the referees for their careful readings and valuable suggestions.

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Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsAhi Evran UniversityKirsehirTurkey
  2. 2.Department of MathematicsBilkent UniversityAnkaraTurkey
  3. 3.Department of MathematicsAnkara UniversityAnkaraTurkey
  4. 4.Department of MathematicsHacettepe UniversityAnkaraTurkey

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