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Abstract
It is shown that if e is an idempotent in a ring R such that both eRe and (1 − e)R(1 − e) are clean rings, then R is a clean ring. This implies that the matrix ring M n (R) over a clean ring is clean, and it gives a quick proof that every semiperfect is clean. Other extensions of clean rings are studied, including group rings.
ACKNOWLEDGMENT
This research was supported by N.S.E.R.C Grant A8075. We would like to thank Professor J.K. Park for reading the paper and making helpful suggestions.