A major part of this paper is devoted to an in-depth study of j-operators and their properties.
This study enables us to obtain several results on liftings and weak liftings of DG modules
along simple extensions of DG algebras and unify the proofs of the existing results obtained
by the authors on these subjects. Finally, we provide a new characterization of the (weak)
lifting property of DG modules along simple extensions of DG algebras.