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.
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Han et al. (2001) studied this question.
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