Cofinitely ⊕-essential supplemented modules

In this paper, cofinitely ⊕-essential supplemented (cofinitely ⊕-e-supplemented) modules are defined and some properties of these modules are investigated. We prove that the finite direct sum of cofinitely ⊕-essential supplemented modules is also cofinitely ⊕-essential supplemented. Let [Formula: see text] be a cofinitely ⊕-essential supplemented an [Formula: see text]-module and let [Formula: see text]. Then we prove that if [Formula: see text] is a direct summand of [Formula: see text] for every direct summand [Formula: see text] of [Formula: see text], then [Formula: see text] is cofinitely ⊕-e-supplemented. Let [Formula: see text] be a distributive and cofinitely ⊕-essential supplemented [Formula: see text]-module. Lastly we prove that [Formula: see text] has no proper cofinitely ⊕-essential submodule.