On a variation of \(\oplus\)-supplemented modules (Q6630965)

This paper studies a generalisation of semi-simplicity of modules over a unital ring \(R\). A left \(R\)-module \(M\) is \textit{\(\oplus_{ss}\)-supplemented} if for every submodule \(U, \ M\) has a direct summand \(V\) such that \(M= U+V\) and \(U\cap V \subseteq \mathrm{Soc}_s(V)\), where \(\mathrm{Soc}_s(V)\) is the sum of all simple submodules that are small in \(V\).\N\NThe author explores the properties of \(\oplus_{ss}\)-supplemented modules \(M\) and relates them to other generalisations of semi-simple modules in the literature. For example, the Jacobson radical of \(M\) is semisimple and the socle of \(M\) is essential in \(M\). \(R\) is a semiperfect ring with semisimple radical if and only if every free left module is \(\oplus_{ss}\)-supplemented. A commutative ring \(R\) is an artinian serial ring with semisimple radical if and only if every R-module is \(\oplus_{ss}\)-supplemented.