Strongly Clean Property and Stable Range One of Some Rings

Abstract

Let R be an associative ring with identity. An element a ∈ R is called clean if a = e + u with e an idempotent and u a unit of R, and a is called strongly clean if, in addition, eu = ue. A ring R is clean if every element of R is clean, and R is strongly clean if every element of R is strongly clean. When is a matrix ring over a strongly clean ring strongly clean? Does a strongly clean ring have stable range one? For these open questions, we prove that 𝕄 _n (C(X)) is strongly π-regular (hence, strongly clean) where C(X) is the ring of all real valued continuous functions on X with X a P-space; C(X) is clean iff it has stable range one; and a unital C*-algebra in which every unit element is self-adjoint is clean iff it has stable range one. The criteria for the ring of complex valued continuous functions C(X,ℂ) to be strongly clean is given.

Key Words:

• C*-algebra
• Clean ring
• Matrix ring
• Ring of continuous functions C(X)
• Ring of continuous functions C(X
• ℂ)
• Stable range one
• Strongly clean ring

2000 Mathematics Subject Classification:

• Primary 16U99
• Secondary 54C30, 46L05

ACKNOWLEDGMENTS

The first author was grateful to Professor Jie Xiao for reading an earlier version of the manuscript. 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 S. Sehgal.