Rad-⊕-Supplemented Semimodules over Semirings

. In this paper, Rad-⊕-supplemented semimodules are defined as generalization of ⊕-supplemented semimodules. Let R be a semiring. An R-semimodule A is called a Rad-⊕-supplemented semimodule, if each subsemimodule of A has a Rad-supplement which is a direct summand of A. Here, we investigate some properties of these semimodules and generalize some results on Rad-⊕-supplemented modules to semimodules. We prove that any finite direct sum of Rad-⊕-supplemented semimodules is Rad-⊕-supplemented. Also, we prove that if A is a subtractive semimodule with (D3) then A is Rad-⊕-supplemented if and only if every direct summand to A is Rad-⊕-supplemented.