On modules that complement direct summands (Q1082407)

An R-module M is said to ''complement direct summands'' if for every direct summand B of M and for every decomposition \(M=\oplus_{i\in I}A_ i\), where every \(A_ i\) is completely incomposable, there exists a subset K of the index set I with \(M=B\oplus (\oplus_{k\in K}A_ k)\). The following characterizations are mentioned by \textit{M. Harada} [Publ. Dépt. Math., Lyon 11, No.2, 19-104 (1974; Zbl 0335.13006)]: Let \(M=\oplus_{i\in I}A_ i\) be a completely indecomposable decomposition; then the following properties are equivalent: (1) M satisfies the take- out property; (2) Any direct summand of M has the exchange property in M; (3) M complements direct summands; (4) \((A_ i: I)\) is a locally-semi-T- nilpotent family; (5) J'\(\cap End(M)\) is the Jacobson radical of End(M). In this note an alternative and elementary proof of the step ''(4)\(\Rightarrow (5)''\) is presented.