Modules whose cofinite closed submodules have weak Rad-supplements

In this paper, we study cofinitely closed weak Rad-supplemented modules (abbreviated as [Formula: see text] modules) serving as proper generalization encompassing the weak Rad-supplemented modules (abbreviated as [Formula: see text] modules) and closed weak Rad-supplemented modules (abbreviated as [Formula: see text] modules). However, we proved that for a finitely generated semisimple [Formula: see text]-module [Formula: see text], [Formula: see text] coincides with [Formula: see text]. Additionally, we proved that a module [Formula: see text] is [Formula: see text]-supplemented if and only if [Formula: see text] is a [Formula: see text] module under certain special assertions. Several results are investigated to establish the relation between variants of supplemented modules such as ⊕-supplemented, Rad-supplemented, [Formula: see text] and [Formula: see text]-supplemented. Further, we studied the property of [Formula: see text] module over several rings such as right nonsingular rings, right nonsingular V-rings, semiperfect ring and Dedekind domain. Finally, we proved that if a finitely generated module [Formula: see text] over Dedekind domain [Formula: see text] is [Formula: see text] then torsion submodule [Formula: see text] and quotient module [Formula: see text] are also [Formula: see text].