Communications of the
Korean Mathematical Society CKMS

Let $R$ be a commutative ring with identity. Let $R$ be an integral domain and $M$ a torsion-free $R$-module. We investigate the relation between the notion of e-exactness, recently introduced by Akray and Zebari \cite{a a}, and generalized the concept of homology, and establish a relation between e-exact sequences and homology of modules. We modify some applications of e-exact sequences in homology and reprove some results of homology with e-exact sequences such as horseshoe lemma, long exact sequences, connecting homomorphisms and etc. Next, we generalize two special drived functor $Tor$ and $Ext$, and study some properties of them.

Keywords: E-exact sequence, e-exact functor, homology, drived functor, $Tor$, $Ext$

MSC numbers: Primary 46M18; Secondary 13C10, 13C12