In this article, we give a generalization to injective modules by using
e-exact sequences introduced by Akray in [1] and name it e-injective
modules and investigate their properties. We reprove both Baer criterion and
comparison theorem of homology using e-injective modules and e-injective
resolutions. Furthermore, we apply the notion e-injective modules into local
cohomology to construct a new form of the cohomology modules call it essential
cohomology modules (briefly e-cohomology modules). We show that the torsion
functor Ξaβ(β) is an e-exact functor on torsion-free modules. We
seek about the relationship of e-cohomology within the classical cohomology.
Finally, we conclude that they are different on the vanishing of their ithβ
cohomology modules