# On Strongly \(J_n\)-Clean Rings

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## Abstract

A ring *R* is said to be strongly \(J_n\)-clean if *n* is the least positive integer such that every element *a* is strongly \(J_n\)-clean, that is, there exists an idempotent *e* such that \(ea=ae\), \(a-e\) is a unit and \(ea^n\) is in the Jacobson radical *J*(*R*). It is proved that strongly \(J_n\)-clean ring is a strongly clean ring with stable range one. If *R* is an abelian ring (a ring in which all idempotents are central), then *R* is strongly \(J_n\)-clean if and only if *R* / *J*(*R*) is strongly \(\pi \)-regular and idempotents lift modulo *J*(*R*). Some examples and basic properties of these rings are studied. Some criterions in terms of solvability of the characteristic equation are obtained for such a \(2\times 2\) matrix to be strongly \(J_2\)-clean.

## Keywords

Strongly \(J_n\)-clean ring Strongly \(\pi \)-rad clean Strongly \(\pi \)-regular ring Strongly clean ring## Mathematics Subject Classification

16U99 16S50## Notes

### Acknowledgements

The research was supported by the National Natural Science Foundation of China (no. 11226071), the Jiangsu Government Scholarship for Overseas Studies and the Grant of Nanjing University of Posts and Telecommunications (no. NY213183).

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