On Quasi-Exact Sequences

On Left Quasi-Exact Sequences

“…Generalizing exact sequences to quasi-exact sequences gives possibilities to generalized some related notions which are defined by exact sequences approach. In this work, we continue to observe further properties of quasi-exact sequences introduced by Anvariyeh and Davvaz ([2] and [3]). We will restrict our discussion to left quasi-exact sequences as a generalization of left exact sequences.…”

“…The generalization of Snake Lemma was observed by Davvaz and Solt in quasi-exact sequences and they gave some results in concepts of generalization in algebra homology [4]. Furthermore, Anvariyeh and Davvaz investigated some properties of finitely generated modules, essential submodules, small submodules and Schanuel Lemma in a short U -exact sequence [3].…”

“…is called a short U -exact sequence or short quasi-exact sequence if f is injective, g is surjective and Im(f ) = g −1 (U ) for some submodule U in C [3]. Anvariyeh and Davvaz studied some properties of short U -exact sequence such as the generalization of Five Lemma, ascending and descending chain of modules in sequences [5].…”

On X-sub-linearly independent modules

“…Then, Anvariyeh dan Davvaz [7] proved further results about quasi-exact sequences and introduced generalization of Schanuel Lemma. Moreover, they obtained some relationships between quasi-exact sequences and superfluous (or essential) submodules.…”

“…They gave a generalization of the Lambek Lemma, Snake Lemma, connecting homomorphism and exact triangle and they established new basic properties of the U -homological algebra. In [8], Anvariyeh and Davvaz studied U -split sequences and established several connections between U -split sequences and projective modules. Let K, L, M be R-modules and X a submodule of L. The triple (K, L, M ) is said to be an…”