One of the useful generalization of extending notion is $FI$-extending property. A module is called $FI$-extending if every fully invariant submodule is essential in a direct summand. In this paper, we explore Weak $FI$-extending concept by considering only semisimple fully invariant submodules rather than all fully invariant submodules. To this end, we call such a module Weak $FI$-extending. We obtain that $FI$-extending modules are properly contained in this new class of modules. Amongst other structural properties, we also deal with direct sums and direct summands of Weak $FI$-extending modules.
"Modules in which semisimple fully invariant submodules are essential in summands,"
Turkish Journal of Mathematics: Vol. 43:
5, Article 20.
Available at: https://journals.tubitak.gov.tr/math/vol43/iss5/20