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Weak clean index of a ring

  • Dhiren Kumar BasnetEmail author
  • Jayanta Bhattacharyya
Original Paper
  • 5 Downloads

Abstract

Motivated by the concept of clean index of rings (Lee and Zhou in Commun Algebra 40:807–822, 2012), we introduce the concept of weak clean index of rings. For any element a of a ring R with unity, we define \( \chi (a)=\{e\in R\mid e^2=e\text { and }a-e \text{ or } a+e \text{ is } \text{ a } \text{ unit }\}\). The weak clean index of R is defined as \(\sup \{|\chi (a)|: a\in R\}\) and it is denoted by \({{\,\mathrm{Win}\,}}(R),\) where \(| \chi (a)| \) denotes the cardinality of the set \(\chi (a)\). In this article, we characterize rings of weak clean indices 1, 2 and 3.

Keywords

Clean ring Weak clean ring Clean index Weak clean index 

Mathematics Subject Classification

16N40 16U99 

Notes

Acknowledgements

The authors would like to thank the referee for his/her careful reading and useful suggestions.

References

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  4. Lee, T.K., Zhou, Y.: Rings of clean index 4 and application. Commun. Algebra 41, 238–259 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Nicholson, W.K.: Lifting idempotents and exchange rings(\(^1\)). Trans. Am. Math. Soc. 229, 269–278 (1977)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Managing Editors 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTezpur UniversityTezpurIndia

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