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Czechoslovak Mathematical Journal

, Volume 69, Issue 1, pp 197–205 | Cite as

Nil-Clean and Unit-Regular Elements in Certain Subrings of \(\mathbb{M}_{2}(\mathbb{Z})\)

  • Yansheng WuEmail author
  • Gaohua Tang
  • Guixin Deng
  • Yiqiang Zhou
Article
  • 62 Downloads

Abstract

An element in a ring is clean (or, unit-regular) if it is the sum (or, the product) of an idempotent and a unit, and is nil-clean if it is the sum of an idempotent and a nilpotent. Firstly, we show that Jacobson’s lemma does not hold for nil-clean elements in a ring, answering a question posed by Koşan, Wang and Zhou (2016). Secondly, we present new counter-examples to Diesl’s question whether a nil-clean element is clean in a ring. Lastly, we give new examples of unit-regular elements that are not clean in a ring. The rings under consideration in our examples are particular subrings of \(\mathbb{M}_{2}(\mathbb{Z})\).

Keywords

clean element nil-clean element unit-regular element Jacobson’s lemma for nil-clean elements 

MSC 2010

16U60 16S50 11D09 

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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

  • Yansheng Wu
    • 1
    Email author
  • Gaohua Tang
    • 2
    • 3
    • 4
  • Guixin Deng
    • 5
  • Yiqiang Zhou
    • 6
  1. 1.Department of MathematicsNanjing University of Aeronautics and AstronauticsNanjing, JiangsuP.R. China
  2. 2.Key Laboratory of Environment Change and Resources Use in Beibu Gulf (Guangxi Teachers Education University)Ministry of EducationGuangxiP.R. China
  3. 3.School of Mathematics and StatisticsGuangxi Teachers Education UniversityNanning, GuangxiP.R. China
  4. 4.School of SciencesQinzhou UniversityQinzhou, GuangxiP.R. China
  5. 5.School of Mathematics and StatisticsGuangxi Teachers Education UniversityNanning, GuangxiP.R. China
  6. 6.Department of Mathematics and StatisticsMemorial University of NewfoundlandSt. John’sCanada

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