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The notion of U-exact sequence in modules over rings was introduced in [20] as a generalization of {0}-exact sequence. In this paper, we prove further results on U-exact and V -coexact sequences where V is induced by U. As shown in the commutative diagram fig. 1, wherein if row-1 is U-exact and row-2 is U'-exact, then we prove that the sequence (0) → ker f'→ ker f → ker f" is (ker f"∩U)-exact, and the sequence (0) → coker f'→ coker f → coker f" is (U' +Im f"/Im f")-exact. We provide explicit examples of the existence of U-exact and U'-coexactness. [ABSTRACT FROM AUTHOR]

In this paper, the (Co)homology theory of pure algebra will be studied. We aim to study the homological theory of cyclic homology and its properties. The triviality property of cohomology groups of algebras will be obtained. We will study the Mayer-Vietortis of cyclic homology. The long exact sequence of cyclic homology and the relation between cyclic homology and dihedral homology will be proved. We will introduce the proof of isomorphism HCn(A×A′)≅HCn(A)⊕HCn(A′) and the definition of Morita invariance for cyclic homology.

The objective of this study is to calculate the K-Cohomology invariants of certain classification spaces of homotopy groups in short exact sequences, where the computation is done using properties of the K-theoretical features. For these packages we use various applications including K-theory Algebra, which transformed the theory of operator algebra from a smaller branch of functional analysis into a topic with deep and wide application areas in all algebra and topology in mathematics.

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]

We prove that the category of F-coalgebras admits exact sequences, under condition F preserves pullbacks; the underlying category 풞 is assumed to be cocomplete with epi-(regular mono) factorizations, finite products and exact sequences. [ABSTRACT FROM AUTHOR]

Finite element methods developed for unfitted meshes have been widely applied to various interface problems. However, many of them resort to non-conforming spaces for approximation, which is a critical obstacle for the extension to $\textbf{H}(\text{curl})$ equations. This essential issue stems from the underlying Sobolev space $\textbf{H}^s(\text{curl};\,\Omega)$ , and even the widely used penalty methodology may not yield the optimal convergence rate. One promising approach to circumvent this issue is to use a conforming test function space, which motivates us to develop a Petrov–Galerkin immersed finite element (PG-IFE) method for $\textbf{H}(\text{curl})$ -elliptic interface problems. We establish the Nédélec-type IFE spaces and develop some important properties including their edge degrees of freedom, an exact sequence relating to the $H^1$ IFE space and optimal approximation capabilities. We analyse the inf-sup condition under certain assumptions and show the optimal convergence rate, which is also validated by numerical experiments. [ABSTRACT FROM AUTHOR]

We prove a number of results on the survival of the type-I property under extensions of locally compact groups: (a) that given a closed normal embedding \mathbb {N}\trianglelefteq \mathbb {E} of locally compact groups and a twisted action (\alpha,\tau) thereof on a (post)liminal C^*-algebra A the twisted crossed product A\rtimes _{\alpha,\tau }\mathbb {E} is again (post)liminal and (b) a number of converses to the effect that under various conditions a normal, closed, cocompact subgroup \mathbb {N}\trianglelefteq \mathbb {E} is type-I as soon as \mathbb {E} is. This happens for instance if \mathbb {N} is discrete and \mathbb {E} is Lie, or if \mathbb {N} is finitely-generated discrete (with no further restrictions except cocompactness). Examples show that there is not much scope for dropping these conditions. In the same spirit, call a locally compact group \mathbb {G} type-I-preserving if all semidirect products \mathbb {N}\rtimes \mathbb {G} are type-I as soon as \mathbb {N} is, and linearly type-I-preserving if the same conclusion holds for semidirect products V\rtimes \mathbb {G} arising from finite-dimensional \mathbb {G}-representations. We characterize the (linearly) type-I-preserving groups that are (1) discrete-by-compact-Lie, (2) nilpotent, or (3) solvable Lie. [ABSTRACT FROM AUTHOR]

In this paper we define the concepts of a "strict morphism" as a suitable map between geometric spaces (as a generalization of hyperstructures), and its kernel. Then we introduce the concepts of a "quotient geometric space", an "exact sequence", an "exact hypersequence", a "homology" and a "hyperhomology" of geometric spaces. Finally we generalize some famous theorems from groups and modules in this direction, such as the "isomorphism theorem", the "five short lemma", the "snake lemma", the "exact triangle", etc., by some conditions that are naturally valid for groups. [ABSTRACT FROM AUTHOR]

The main objective of the research is to study the first natural triangular representation of the symmetric groups over a field K of characteristic p≠ 2 which deals with the partition λ = (n − 4,3,1) of the positive integer n. Furthermore, this work has proven that the S(n − 4,3,1) is a submodule of F1 . The F1 = KSn(x2x4x6x7² −x2x3x6x7² − x2x4x5x7² + x²x³x5x7² ) can be only split whenp|(n − 5). الهدف من هذا البحث هو دراسة [ABSTRACT FROM AUTHOR]

The notion of a U-exact sequence is a generalization of the exact sequence. In this paper, we introduce a rough U-exact sequence in a rough group in an approximation space. Furthermore, we provide the properties of the rough U-exact sequence in a rough group.

A (right) module bundle over an algebra bundle ξ = (ξ; p;X) is a vector bundle M=(M,q,X) together with a morphism Ф: M ⊕ ξ-M which induces a right ξx-module structure onMx for each x ∊ X. In this paper, we prove some important results involving exact sequences of ξ-module bundles, mainly, the snake lemma. [ABSTRACT FROM AUTHOR]

Bousfield introduced an algebraic category of modules that reflects the structure detected by p-localized complex topological K-theory. He constructed, for any module M in this category, a natural 4-term exact sequence 0 → M → U M → U M → M ⊗ Q → 0 , where U denotes the co-free functor, right adjoint to the forgetful functor to Z (p) -modules. Clarke et al. identified the objects of Bousfield's category as the 'discrete' modules for a certain topological ring A, obtained as a completion of the polynomial ring Z (p) [ x ] , and simplified the construction of the Bousfield sequence in this context. We introduce the notion of 'cobasically discrete' R-modules as a clarification of the Clarke et al. modules, noting that these correspond to comodules over the coalgebra that R is dual to. We study analogues of the Bousfield sequence for other polynomial completion rings, noting a variety of behaviour in the last term of the sequence. [ABSTRACT FROM AUTHOR]

An extension of X by Y is a short exact sequence of quasi Banach modules and homomorphisms 0 -→ Y -→ Z -→ X -→ 0. When properly organized all these extensions constitute a linear space denoted by ExtB(X, Y ), where B is the underlying (Banach) algebra. In this paper we "compute" the spaces of extensions for the Schatten classes when they are regarded in its natural (left) module structure over B = B(H), the algebra of all operators on the ground Hilbert space. Our main results can be summarized as follows: ... In the first case, every extension 0 -→ Sq -→ Z -→ Sp -→ 0 splits and so X = Sq -Sp. In the second case, every self-extension of Sp arises (and gives rise) to a minimal extension of S1 in the quasi Banach category, that is, a short exact sequence 0 -→ C -→ M -→ S1 -→ 0. In the third case, each extension corresponds to a "twisted Hilbert space", that is, a short exact sequence 0 -→ H -→ T -→ H -→ 0. Thus, the subject of the paper is closely connected to the early "three-space" problems studied (and solved) in the seventies by Enflo, Lindenstrauss, Pisier, Kalton, Peck, Ribe, Roberts, and others. [ABSTRACT FROM AUTHOR]

In this paper we calculate cohomology of a classical Lie algebra of type A 2 over an algebraically field k of characteristic p = 3 with coefficients in simple modules. To describe their structure we will consider them as modules over an algebraic group SL 3( k ) . In the case of characteristic p = 3 , there are only two peculiar simple modules: a simple that module isomorphic to the quotient module of the adjoint module by the center, and a one-dimensional trivial module. The results on the cohomology of simple nontrivial module are used for calculating the cohomology of the adjoint module. We also calculate cohomology of the simple quotient algebra Lie of A 2 by the center.