C1-modules and generalizations

R as a ring, a module M defined on R is called C1 (CS or extending) moduleif every submodule of M is essential in a direct summand. It can be expressedas equivalence of this definition that every complement submodule of M is adirect summand of M. There are various generalizations for C1-module family.C11-modules and FI-extending modules are two of them. A module M is aC11-module if every submodule has a complement which is a direct summandof M. A module M is called FI-extending if every fully invariant submoduleis essential in a direct summand. Clearly, FI-extending modules are one of thegeneralization of C11-modules. In this study, basic properties of these threemodule families are examined and the relationship among them is studied.