A Note on Strongly Clean Matrix Rings

Abstract

Let R be an associative ring with identity. An element a ∈ R is called strongly clean if a = e + u with e ² = e ∈ R, u a unit of R, and eu = ue. A ring R is called strongly clean if every element of R is strongly clean. Strongly clean rings were introduced by Nicholson [7]. It is unknown yet when a matrix ring over a strongly clean ring is strongly clean. Several articles discussed this topic when R is local or strongly π-regular. In this note, necessary conditions for the matrix ring 𝕄 _n (R) (n > 1) over an arbitrary ring R to be strongly clean are given, and the strongly clean property of 𝕄₂(RC ₂) over the group ring RC ₂ with R local is obtained.

Key Words:

• Group ring
• Local ring
• Matrix ring
• Strongly clean ring

2000 Mathematics Subject Classification:

• 16U60
• 16S34

ACKNOWLEDGMENTS

The authors were grateful to Professor E. Sánchez Campos for reading an earlier version of the manuscript and kindly permitting them to include the proof of Theorem 2.5. The first author was supported by NSERC, Canada, and the second was supported by Initial Grant of Harbin Institute of Technology, China.

Notes

Communicated by M. Cohen.