cofinitely ⨁dj-supplemented modules

For any r-module w, dj (w) presented as the total of all j-small sub-modules. if a and b are sub-module of w, we say a is ∩ dj ’supplement of b in w if w = a+b = a ⨁ á, for á →w, and a ∩ b ≪j dj (a). if every sub-module has ⨁ dj -supplemented, then w is ⨁ dj -supplemented a sub-module a of w. if a sentence is conclusive, it is said to be cofinite i.e., w/a is finitely generated. also we introduce cofinite ⨁ dj -supplemented if every cofinite sub-module of w has ⨁ dj -supplemented.