The homology regarding to e-exact sequences
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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.
Abstract
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.
참고문헌 (4)
- Ismael Akray, "Essential exact sequences", 대한수학회논문집, 35(2), pp.469~480 (2020), 대한수학회[DOI]
- F. W. Anderson, "Rings and Categories of Modules, Graduate Texts in Mathematics, 13" (1992), Springer-Verlag
- J. J. Rotman, "An Introduction to Homological Algebra, Pure and Applied Mathematics, 85" (1979), Academic Press, Inc.
- C. A. Weibel, "History of homological algebra, in History of topology", pp.797~836 (1999), North-Holland