Rings whose modules are \(\oplus\)-supplemented
A module \(M\) is called lifting (or satisfies (D1)) if for every submodule \(N\) of \(M\), there are summands \(K\) and \(K'\) of \(M\) such that \(M=K\oplus K'\), \(K'\) is contained in \(N\), and \(N\cap K\) is small in \(K\). Lifting modules have been studied extensively, e.g., in the monograph of \textit{S. H. Mohamed} and \textit{B. J. Mueller} [Continuous and discrete modules, Lond. Math. Soc. Lect. Note Ser. 147, Cambridge Univ. Press, Cambridge (1990; Zbl 0701.16001)]. Generalizing this concept, a module \(M\) is defined to be \(\oplus\)-supplemented if for every submodule \(N\) of \(M\) there is a summand \(K\) of \(M\) such that \(M=N+K\) and \(N\cap K\) is small in \(K\). The authors give several interesting characterizations of rings in terms of \(\oplus\)-supplemented modules. It is shown that every finitely generated right \(R\)-module is \(\oplus\)-supplemented if and only if every cyclic right \(R\)-module is \(\oplus\)-supplemented and every finitely generated right \(R\)-module is a direct sum of cyclic modules. Also, a ring \(R\) is serial if and only if every finitely presented right and finitely presented left \(R\)-module is \(\oplus\)-supplemented. Moreover, every finitely generated right and finitely generated left \(R\)-module is \(\oplus\)-supplemented if and only if \(R\) is serial and every indecomposable injective right and left \(R\)-module is uniserial. Finally, it is proved that every right \(R\)-module and left \(R\)-module is \(\oplus\)-supplemented if and only if \(R\) is Artinian serial.
scientific article; zbMATH DE number 1348167
scientific article