It is well known that the ring radical theory can be approached via language of modules. In this work, we present some generalizations of classical results from module theory, in the two-sided and graded sense. Let G be a group, F an algebraically closed field with char ( F ) = 0 , A a finite dimensional G-graded associative F -algebra and M a G-graded unitary A -bimodule. We proved that if A = M n ( F σ [ H ] ) with an elementary-canonical G-grading, where H is a finite abelian subgroup of G and σ ∈ Z 2 ( H , F * ) , then M being irreducible graded implies that there exists a nonzero homogeneous element w ∈ M satisfying M = B w and B w = w B . Another result we proved generalizes the last one: if G is abelian, A is simple graded and M is finitely generated, then there exist nonzero homogeneous elements w 1 , w 2 , … , w n ∈ M such that M = A w 1 ⊕ A w 2 ⊕ ⋯ ⊕ A w n , where w i A = A w i ≠ 0 for all i = 1 , 2 , … , n , and each A w i is irreducible. The elements wi ’s are associated with the irreducible characters of G. We also describe graded bimodules over graded semisimple algebras. And we finish by presenting a Pierce decomposition of the graded Jacobson radical of any finite dimensional F -algebra with a G-grading.