  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.
