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


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 \(\mathbb{M}_{2}(\mathbb{Z})\).


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

MSC 2010

16U60 16S50 11D09 


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  1. [1]
    D. Andrica, G. Călugăreanu: A nil-clean 2×2 matrix over the integers which is not clean. J. Algebra Appl. 13 (2014), Article ID 1450009, 9 pages.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    V. P. Camillo, D. Khurana: A characterization of unit regular rings, Commun. Algebra. 29 (2001), 2293–2295.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    J. Chen, X. Yang, Y. Zhou: On strongly clean matrix and triangular matrix rings, Commun. Algebra. 34 (2006), 3659–3674.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    D. Cvetkovic-Ilic, R. Harte: On Jacobson’s lemma and Drazin invertibility, Appl. Math. Lett. 23 (2010), 417–420.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    A. J. Diesl: Nil clean rings, J. Algebra. 383 (2013), 197–211.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    D. Khurana, T. Y. Lam: Clean matrices and unit-regular matrices, J. Algebra. 280 (2004), 683–698.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    T. Koşan, Z. Wang, Y. Zhou: Nil-clean and strongly nil-clean rings, J. Pure Appl. Algebra. 220 (2016), 633–646.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    T. Y. Lam, P. P. Nielsen: Jacobson’s lemma for Drazin inverses. Ring Theory and Its Applications (D. V. Huynh et al., eds.). Contemporary Mathematics 609, American Mathematical Society, Providence, 2014, pp. 185–195.Google Scholar
  9. [9]
    G. Tang, Y. Zhou: A class of formal matrix rings, Linear Algebra Appl. 438 (2013), 4672–4688.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    G. Zhuang, J. Chen, J. Cui: Jacobson’s lemma for the generalized Drazin inverse, Linear Algebra Appl. 436 (2012), 742–746.MathSciNetCrossRefzbMATHGoogle Scholar

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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