EXTENSIONS OF CLEAN RINGS

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.