In this paper, we introduce the notion of F-CS-Rickart modules, where F is a given fully invariant submodule. These modules are a generalization of CS-Rickart modules and F-inverse split modules. We characterize F-CS-Rickart modules and investigate several properties of F-CS-Rickart modules. We show that any F-CS-Rickart module can be written as a direct sum of two submodules, one of which is an essential extension of F and the other of which is a CS-Rickart module. In addition, we show that any image of an F-CS-Rickart projective module satisfying C₂ condition can be written as a direct sum of two submodules, one of which is a projective module and the other of which is contained in F^*.

F-CS-Rickart module, F-inverse split module, CS-Rickart module, Rickart module.