On Generalized FI-extending Modules

A module M is called FI-extending if every fully invariant submodule of M is essential in a direct summand of M . In this work, we define a module M to be generalized FI-extending (GFI-extending) if for any fully invariant submodule N of M , there exists a direct summand D of M such that N ≤ D and that D/N is singular. The classes of FI-extending modules and singular modules are properly contained in the class of GFIextending modules. We first develop basic properties of this newly defined class of modules in the general module setting. Then, the GFI-extending property is shown to carry over to matrix rings. Finally, we show that the class of GFI-extending modules is closed under direct sums but not under direct summands. However, it is proved that direct summands are GFI-extending under certain restrictions.