In this thesis, we call a module FI-extending if every fully invariant submodule is essential in a direct summand. Initially we develop basic properties in the module setting. For example in contrast to extending modules, a direct sum of FI-extending modules is FI-extending (but direct product of FI-extending modules need not be FI-extending). Later we largely focus on the specific case when a ring is FI-extending (considered as a module over itself).