Advertisement

On Medium *-Clean Rings

  • Huanyin Chen
  • Marjan Sheibani Abdolyousefi
  • Handan Kose
Article
  • 12 Downloads

Abstract

A *-ring R is called a medium *-clean ring if every element in R is the sum or difference of an element in its Jacobson radical and a projection that commute. We prove that a ring R is medium *-clean if and only if R is strongly *-clean and R / J(R) is a Boolean ring, \({\mathbb {Z}}_3\) or the product of such rings, if and only if R weakly J-*-clean and \(a^2\in R\) is uniquely *-clean for all \(a\in R\), if and only if every idempotent lifts modulo J(R), R is abelian and R / J(R) weakly *-Boolean. A subclass of medium *-clean rings with many nilpotents is thereby characterized.

Keywords

Projection Jacobson radical homomorphic image *-clean ring 

Mathematics Subject Classification

Primary 16W10 Secondary 16E50 

Notes

Acknowledgements

The authors would like to thank the referee for his/her careful reading and valuable remarks that improved the presentation of our work. H. Chen was supported by the Natural Science Foundation of Zhejiang Province, China (no. LY17A010018).

References

  1. 1.
    Breaz, S., Danchev, P., Zhou, Y.: Rings in which every element in either a sum or a difference of a nilpotent and an idempotent. J. Algebra Appl. 15, 1650148 (2016).  https://doi.org/10.1142/S0219498816501486
  2. 2.
    Berberian, S.K.: Baer *-Rings. Springer, Heidelberg (2011)zbMATHGoogle Scholar
  3. 3.
    Chen, H.: Rings Related Stable Range Conditions, Series in Algebra 11. World Scientific, Hackensack (2011)CrossRefGoogle Scholar
  4. 4.
    Chen, H., Harmanci, A., Özcan, A.C.: Strongly J-clean rings with involutions. Contemp Math 609, 33–44 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, H., Sheibani, M.: Strongly 2-nil-clean rings. J. Algebra Appl. 16, 1750178 (2017).  https://doi.org/10.1142/S021949881750178X MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cui, J., Wang, Z.: A note on strongly *-clean rings. J. Korean Math. Soc. 52, 839–851 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hirano, Y., Tominaga, H.: Rings in which every element is a sum of two idempotents. Bull. Austral. Math. Soc. 37, 161–164 (1988)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kosan, M.T., Zhou, Y.: On weakly nil-clean rings. Front. Math. China (2016).  https://doi.org/10.1007/s11464-016-0555-6
  9. 9.
    Li, C., Zhou, Y.: On strongly *-clean rings. J. Algebra Appl. 10, 1363–1370 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Li, Y., Parmenter, M.M., Yuan, P.: On *-clean group rings. J. Algebra Appl. 14, 1550004 (2015).  https://doi.org/10.1142/S0219498815500048 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Stancu, A.: A note on commutative weakly nil clean rings. J. Algebra Appl. 15, 1620001 (2016).  https://doi.org/10.1142/S0219498816200012 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Vas, L.: *-Clean rings; some clean and almost clean Baer *-rings and von Neumann algebras. J. Algebra 324, 3388–3400 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Yu, H.P.: On quasi-duo rings. Glasg. Math. J. 37, 21–31 (1995)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Huanyin Chen
    • 1
  • Marjan Sheibani Abdolyousefi
    • 2
  • Handan Kose
    • 3
  1. 1.Department of MathematicsHangzhou Normal UniversityHangzhouChina
  2. 2.Women’s University of Semnan (Farzanegan)SemnanIran
  3. 3.Department of MathematicsAhi Evran UniversityKirsehirTurkey

Personalised recommendations