Commun. Korean Math. Soc. Published online January 10, 2020

Ismael Akray and Amin Zebari
Soran university

Abstract : Let $R$ be a commutative ring with identity and $M$ a unital $R-$module. We give a new generalization to exact sequences called e-exact sequence. A sequence $ 0 \rightarrow A \stackrel{f}{\rightarrow} B \stackrel{g}{\rightarrow} C \rightarrow 0$ is said to be e-exact if $f$ is monic, $Imf \leq_e Kerg$ and $Img \leq_e C$. We modify many famous theorems including exact sequences to one includes e-exact sequences like $3 \times 3$ lemma, four and five lemmas. Next, we prove that for torsion free module $M$, the contravariant functor $Hom(-, M)$ is left e-exact and the covariant functor $M \otimes -$ is right e-exact. Finally, we define e-projective module and charactrize it. We show that the direct sum of $R-$modules is e-projective module if and only if each summand is e-projective.