Abstract
A ring R is called clean if every element of R is the sum of an idempotent and a unit. Let M be a R-module. It is obtained in this article that the endomorphism ring End(M) is clean if and only if, whenever A = M′ ⊕ B = A1 ⊕ A2 with M′ ≅ M, there is a decomposition M′ =M1 ⊕ M2 such that A = M′ ⊕ [A1 ∩ (M1 ⊕ B)] ⊕ [A2 ∩ (M2 ⊕ B)]. Then unit-regular endomorphism rings are also described by direct decompositions.
ACKNOWLEDGMENTS
The authors are very grateful to the referee for excellent comments and suggestions.