Remarks on the lifting property of simple modules (Q793132)

R is an associative ring with identity and \(\{M_{\alpha}\}_ I\) is an infinite set of completely indecomposable right R-modules. Let \(M=\sum_{I}\oplus M\), \(\bar M=M/J(M)\), where J(M) denotes the Jacobson radical of M. If \(\bar M\) is completely reducible, then M is said to have the lifting property of simple modules modulo the radical if every submodule of \(\bar M\) is induced from a direct summand of M [cf. \textit{M. Harada}, Osaka J. Math. 17, 783-791 (1980; Zbl 0462.16017)]. Now let \(H_ 0\) be the family of all maximal submodules of M. Then M is said to have the lifting property of modules for \(H_ 0\) if there exists a decomposition \(M=A^*\oplus A^{**}\) such that \(A^*\subseteq A\in H_ 0\) and \(A\cap A^{**}\) is small in M [cf. \textit{K. Oshiro}, ibid. 20, 337-372 (1983; Zbl 0516.16015)]. Both concepts are dual to the ''extending property of simple modules'', mentioned in a paper of \textit{M. Harada} and \textit{K. Oshiro} [ibid. 18, 767-785 (1981; Zbl 0473.16018)]. In this paper the authors show: M has the lifting property of modules for \(H_ 0\) if and only if M has the lifting property of simple modules modulo the radical and M satisfies an additional condition on submodules and submodules of \(\bar M\).