On Strongly Clean Modules

An element of a ring R is called “strongly clean” if it is the sum of an idempotent and a unit that commute, and R is called “strongly clean” if every element of R is strongly clean. A module M is called “strongly clean” if its endomorphism ring End(M) is a strongly clean ring. In this article, strongly clean modules are characterized by direct sum decompositions, that is, M is a strongly clean module if and only if whenever M′⊕ B = A ₁⊕ A ₂ with M′? M, there are decompositions M′ = M ₁⊕ M ₂, B = B ₁⊕ B ₂, and A _i  = C _i ⊕ D _i (i = 1,2) such that M ₁⊕ B ₁ = C ₁⊕ D ₂ = M ₁⊕ C ₁ and M ₂⊕ B ₂ = D ₁⊕ C ₂ = M ₂⊕ C ₂.