Kyungpook Mathematical Journal 2020; 60(1): 44-51
Published online March 31, 2020
Copyright © Kyungpook Mathematical Journal.
On Generalized FI-extending Modules
Canan Celep Yücel
Department of Mathematics, Faculty of Arts and Sciences, Pamukkale University, Denizli 20070, Turkey
e-mail : email@example.com
Received: June 3, 2016; Revised: November 2, 2019; Accepted: December 3, 2019
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 GFI-extending 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.
Keywords: fully invariant submodule, FI-extending, GFI-extending.