Fully invariant submodules play an important designation in studying the structure of some known modules such as (dual) Rickart and (dual) Baer modules. In this work, we introduce F-dual Rickart (Baer) modules via the concept of fully invariant submodules. It is shown that M is F-dual Rickart if and only if M = F ⊕L such that F is a dual Rickart module. We prove that a module M is F-dual Baer if and only if M is F-dual Rickart and M has SSSP for direct summands of M contained in F. We present a characterization of right I-dual Baer rings where I is an ideal of R. Some counter-examples are provided to illustrate new concepts.
