Modules having Baer summands

Let R be an arbitrary ring with identity and M a right R-module with S= End(R)(M). Let F be a fully invariant submodule of M and I-1(F) denotes the set {m is an element of M : Im subset of F} for any subset I of S. The module M is called F-Baer if I-1(F) is a direct summand of M for every left ideal I of S. This work is devoted to the investigation of properties of F-Baer modules. We use F-Baer modules to decompose a module into two parts consists of a Baer module and a module determined by fully invariant submodule F, namely, for a module M, we show that M is F-Baer if and only if M = F circle plus N where N is a Baer module. By using F-Baer modules, we obtain some new results for Baer rings.