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Let $R$ be a commutative ring with identity and $M$ a unital $R$-module. We give a new generalization of exact sequences called e-exact sequences. A sequence $ 0 \To A \stackrel{f}{\To} B \stackrel{g}{\To} C \To 0$ is said to be e-exact if $f$ is monic, ${\rm Im}f \leq_e {\rm Ker}g$ and ${\rm Im}g \leq_e C$. 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 ${\rm Hom}(-, M)$ is left e-exact and the covariant functor $M \otimes -$ is right e-exact. Finally, we define e-projective module and characterize it. We show that the direct sum of $R$-modules is e-projective module if and only if each summand is e-projective.
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