⊕-supplemented modules with respect to images of a fully invariant submodule

Lifting modules and their various generalizations as some main concepts in module theory have been studied and investigated extensively in recent decades. Some authors tried to present some homological aspects of lifting modules and ⊕-supplemented modules. In this work, we shall present a homological approach to ⊕-supplemented modules via fully invariant submodules. Lifting modules and H-supplemented modules with respect to images of a fixed fully invariant submodule of a module where investigated in first author’s last works. We intend here to introduce and study a module M such that φ(F) has a supplement as a direct summand of M, for every endomorphism φ of M where F is a fixed fully invariant submodule of M.