Extending property for finitely generated submodules (Q1365442)

Let \(R\) be a ring with identity and let \(M\) be a unital right \(R\)-module. The module \(M\) is extending if every submodule is essential in a direct summand. There is an extensive literature on extending modules [see, for example, \textit{N. V. Dung, D. V. Huynh, R. Wisbauer} and the reviewer, Extending modules (Pitman, London 1994; Zbl 0841.16001)]. In this note the authors define the module \(M\) to be \(ef\)-extending if every closed submodule which is an essential extension of a finitely generated submodule is a direct summand. On the other hand, \(M\) is \(f\)-extending if every finitely generated submodule is essential in a direct summand. The authors obtain some basic properties of \(ef\)-extending and \(f\)-extending modules similar to known results about extending modules.