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Abstract
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.
Notes
Communicated by I. Swanson.