FI-Rad_J-Supplemented and FI-⨁-Rad_J-Supplemented Modules

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Assume that R is any ring with identity and M is a unitary left R-module. This work introduces the Rad_J-Supplemented module with respect to the fully invariant submodule it is denoted by (FI-Rad_J-Supplemented) and ⨁-Rad_J-Supplemented modules with respect to the fully invariant submodule (denoted by FI-⨁-Rad_J-Supplemented modules). As well as the main features of these modules in this work, and various properties of these modules. Also, we defined weakly FI-Rad_J-Supplemented modules as an extension of the FI-Rad_J-supplemented modules and we explain the relationship between them using notes and examples.