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The largest class of multi-valued systems satisfying the module-like axioms are the hyper modules. The main tools concerning the class of Hv -- (R -- S)-bi-modules are fundamental relations. First we define Hv -- (R -- S)-hyper-bi-modules and investigate exact secqenses in this field and find equivalent conditions of exactness. [ABSTRACT FROM AUTHOR]

In this work, we prove by employing mapping Cone that the sequence and the subsequence of the characteristic-zero are exact and subcomplex respectively in the case of partition (6,6,4) . [ABSTRACT FROM AUTHOR]

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In this paper we find a generalization of the snake lemma and Schanuel's lemma in Hv-modules. We define the isomorph sequences and determine the conditions to split the exact sequences in Hv-modules. Some interesting results on these concepts are given. [ABSTRACT FROM AUTHOR]

Quasi module is an axiomatisation of the hyperspace structure based on a module. We proposed this structure in our paper [2]. It can be thought of as a generalisation of the module structure in the sense that every module can be embedded into a quasi module and every quasi module contains a module. The structure of quasi module is designed on a commutative semigroup with an external ring multiplication and a compatible partial order which has a very intrinsic effect in the entire structure and plays a key role in any development of the theory of quasi module. In this paper we have introduced the concept of exact sequence on quasi modules and proved some Theorems related to exact sequences. [ABSTRACT FROM AUTHOR]

The largest class of multivalued systems satisfying the module-like axioms are the Hv -modules. The main tools concerning the class of Hv -modules with the ordinary modules are the fundamental relations. Based on the relation ε* , exact sequences in Hv -modules are defined. In this paper, we introduce the Hv -module M[A] and determine its heart and the connection between equivalence relations ε* M[A] and ε* A . Moreover, we define the M[-] and -[M] functors and investigate the exactness and some concepts related to them. Finally, we prove the five short lemma in Hv -modules. [ABSTRACT FROM AUTHOR]

Let V be a vertex operator algebra. The fusion products in this paper are defined by logarithmic intertwining operators. Under this setting, we prove the pure exactness of any extension of the V -module V . Namely, if 0 → Q → ϵ P → ρ V → 0 is a short exact sequence of V -modules and W is a V -module, then 0 → Q ⊠ W → ϵ ⊠ 1 W P ⊠ W → ρ ⊠ 1 W V ⊠ W → 0 is also exact, where we view it as a sequence of g ( V ) -modules. [ABSTRACT FROM AUTHOR]

Abstract: By means of a mixture of linear and nonlinear techniques, the paper treats quasi-Banach function spaces as -modules and studies the basic algebraical constructions and their interactions, placing the emphasis on tensor products. Sample result: if is an exact sequence of -modules and homomorphisms, where Y and Z are function spaces, and V is another function space, then the tensorized sequence is exact too. [Copyright &y& Elsevier]

Abstract: It is shown that every centralizer from any “metric function space” to is continuous at the origin of . As a consequence, every short exact sequence of -modules splits if is a “minimal” function space, and in particular if . There are pairs of Orlicz function spaces such that , but . [Copyright &y& Elsevier]

Benson and Goodearl [Periodic flat modules, and flat modules for finite groups, Pacific J. Math.196(1) (2000) 45-67] proved that if M is a flat module over a ring R such that there exists an exact sequence of R-modules 0 → M → P → M → 0 with P a projective module, then M is projective. The main purpose of this paper is to generalize this theorem to any exact sequence of the form 0 → M → G → M → 0, where G is an arbitrary module over R. Moreover, we seek counterpart entities in the Gorenstein homological algebra of pure projective and pure injective modules. [ABSTRACT FROM AUTHOR]

The notion of BCK-modules was introduced in [1]. In this paper we establish certain isomorphism theorems of BCK-modules and lay out its applications to the chains of BCK-modules. Moreover, exact sequences of BCK-modules are studied. [ABSTRACT FROM AUTHOR]