D_j -Supplemented Modules

Abstract

Abstract If 𝒜 ≪ j Dj (𝒲), where Dj (𝒲) is the generalised Jacobson radical of 𝒲, then a sub-module 𝒜 of a module 𝒲 is said to be Dj -small. If W = A + B and A n B ≪ j D j ( A ) , 𝒜 is Dj - ’supplement of ℬ in 𝒲. 𝒲 is said to be Dj - ’supplemented if every sub-module of 𝒲 has Dj - ’supplement in 𝒲. If for each 𝒜 a sub-module of 𝒲, there exists a sub-module ℬ such that W = A + B and A n B ≪ j D j ( W ) , then 𝒲 is called Dj -weakly ’supplemented module.