Essential cohomology modules

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 $\Gamma_a ( - )$ 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 $i_{th}$ cohomology modules

Essential ideal transforms

It is our intention in this research generalized some concept in local cohomology such as contravarint functor $ext$, covariant functor $Ext$, covarian functor $Tor$ and ideal transforms with $e$-exact sequences. The $e$-exact sequence was introduced by Akray and Zebari \cite{AZ} in 2020. We obtain for a torsion-free modules $B$, $_eext^n_R(P,B)=0$ while $_eExt^n_R(A,E)=0$ for every module $A$. Also for any torsion-free module $B$ we have an $e$-exact sequence $0\to \Gamma_{a}(B) \to B\to D_{a}(B)\to H^1_{a}(B)\to 0$ and an isomorphisms between $B$ and $r D_{a}(B)$. Finally we generalize Mayer-Vietories with $e$-exact sequences in essential local cohomology, we get a special $e$-exact sequences