Two-sided Two-cosided Hopf Modules and Yetter–Drinfeld Modules for Quasi-Hopf Algebras

For a quasi-Hopf algebra H, an H-bicomodule algebra and an H-bimodule coalgebra C we will show that the category of two-sided two-cosided Hopf modules is equivalent to the category of right–left generalized Yetter–Drinfeld modules . Using alternative versions of this result we will recover the category isomorphism between the categories of left–left and left–right Yetter–Drinfeld modules over a quasi-Hopf algebra.      

Doi–Hopf Modules and Yetter–Drinfeld Modules for Quasi-Hopf Algebras

For a quasi-Hopf algebra H, a left H-comodule algebra  and a right H-module coalgebra C we will characterize the category of Doi–Hopf modules ^C ?(H) in terms of modules. We will also show that for an H-bicomodule algebra  and an H-bimodule coalgebra C the category of generalized Yetter–Drinfeld modules (H) ^C is isomorphic to a certain category of Doi–Hopf modules. Using this isomorphism we will transport the properties from the category of Doi–Hopf modules to the category of generalized Yetter–Drinfeld modules.     

Doi–Hopf Modules Associated to Comodule Coalgebras

To any right comodule coalgebra C over a Hopf algebra H we associate a left H-comodule algebra A. Under certain conditions, in particular in the case where H has nonzero integrals, we show that the category of right C, H-comodules is isomorphic to a certain subcategory of the category of Doi–Hopf modules associated to A. As an application, we investigate the connection between C and the smash coproduct C ? H being right semiperfect.     

Yetter–Drinfel'd Modules for Group-Cograded Multiplier Hopf Algebras

We give a representation-theoretic and a categorical interpretation of the Drinfel'd double into the framework of group-cograded multiplier Hopf algebras. The Drinfel'd double as constructed by Zunino for a finite-type Hopf group-coalgebra is an example of this construction in the sense that the components of the group-cograded multiplier Hopf algebras are unital and finite-dimensional algebras and the admissible action is related with the adjoint action of the group on itself.     

Fundamental Theorems of Weak Doi–Hopf Modules and Semisimple Weak Smash Product Hopf Algebras

Abstract

This paper, mainly gives a Fundamental Theorem of weak Doi–Hopf modules, which is not only generalizes the Fundamental Theorem of weak Hopf modules but also generalizes the Fundamental Theorem of relative Hopf modules. Moreover, it gives a sufficient and necessary condition for weak smash product algebras to be weak bialgebras, and a sufficient condition for weak smash product algebras to be semisimple weak Hopf algebras.     

Breuil–Kisin Modules and Hopf Orders in Cyclic Group Rings

For K, a finite extension of ? _p with ring of integers R, we show how Breuil–Kisin modules can be used to determine Hopf orders in K-Hopf algebras of p-power dimension. We find all cyclic Breuil–Kisin modules and use them to compute all of the Hopf orders in the group ring KΓ where Γ is cyclic of order p or p ². We also give a Laurent series interpretation of the Breuil–Kisin modules that give these Hopf orders.     

Yetter–Drinfeld Modules over Weak Braided Hopf Monoids and Center Categories

In this paper,we introduce several centralizer constructions in a monoidal context and establish a monoidal equivalence with the category of Yetter–Drinfeld modules over a weak braided Hopf monoid.We apply the general result to the calculus of the center in module categories.     

On the Construction of d—Continuous Modules

In this paper we introduce: and study the notion of dual Gontinuous (d-Contmuous) modnles, A decomposrtion thcorem for a d-continuous modules proved, this generahzes all known decompositron theorems for quasi-projective modules.     

Injective Cogenerators,Cotilting Modules and Cosilting Modules

It is well known that the concept of cotilting modules generalizes injective cogenerators and in turn,the concept of cosilting modules generalizes cotilting modules.In this paper,we further investigate the close connections among injective cogenerators,cotilting modules and cosilting modules from the viewpoint of morphism categories.Some applications are also given.     

ω-Gorenstein Modules

We introduce the notion of ω-Gorenstein modules, where ω is a faithfully balanced self-orthogonal module. This gives a common generalization of both Gorenstein projective modules and Gorenstein injective modules. We consider such modules in the tilting theory. Consequently, some results due to Auslander and colleagues and Enochs and colleagues are generalized.     

Meta—projective Modules,Tensor Products and Limits

Inthispaper,allringsconsideredareassociativewithl#o,allmodulesareunital.Meta-projectivemodulesandmeta-injectivemeduleswerefirstintroductedbyChen-Tong[l]andFeng[2j.LetRbearing,andMaleftR-module.M,iscalledamaximalquotientprojectivesubmeduleofMifM,isamaximalsubmoduleandM/M'isprojective.Accordingto[2j,M,iscalledamaximalquotientinjectivesubmoduleofMifMiisamaximalsubmoduleandM/M,isinjective.Missaidtobemeta-projectiveiftheintersectionofMandallmaximal...quotientproJectivemodulesiso.Similarly,M…     

κ-Gorenstein Modules

Let A and F be artin algebras and ∧UГa paper, we first introduce the notion of k-Gorenstein faithfully balanced selforthogonal bimodule. In this modules with respect to ∧UГ and then characterize it in terms of the U-resolution dimension of some special injective modules and the property of the functors Ext^i （Ext^i （-, U）, U） preserving monomorphisms, which develops a classical result of Auslander. As an application, we study the properties of dual modules relative to Gorenstein bimodules. In addition, we give some properties of ∧UГwith finite left or right injective dimension.     

Yetter–Drinfeld Modules over the Hopf–Ore Extension of the Group Algebra of Dihedral Group

Let k be an algebraically closed field of characteristic zero, and D _n be the dihedral group of order 2n, where n is a positive even integer. In this paper, we investigate Yetter-Drinfeld modules over the Hopf-Ore extension A(n, 0) of kD _n . We describe the structures and properties of simple Yetter-Drinfeld modules over A(n, 0), and classify all simple Yetter-Drinfeld modules over A(n, 0).     

Cohen–Macaulay Modules and Holonomic Modules Over Filtered Rings

We study Gorenstein dimension and grade of a module M over a filtered ring whose associated graded ring is a commutative Noetherian ring. An equality or an inequality between these invariants of a filtered module and its associated graded module is the most valuable property for an investigation of filtered rings. We prove an inequality G?dim M ≤ G?dim gr M and an equality grade M = grade gr M, whenever Gorenstein dimension of gr M is finite (Theorems 2.3 and 2.8). We would say that the use of G-dimension adds a new viewpoint for studying filtered rings and modules. We apply these results to a filtered ring with a Cohen–Macaulay or Gorenstein associated graded ring and study a Cohen–Macaulay, perfect, or holonomic module.     

⊕-Cofinitely Supplemented Modules

Let R be a ring and M a right R-module. M is called -cofinitely supplemented if every submodule N of M with M/N finitely generated has a supplement that is a direct summand of M. In this paper various properties of the -cofinitely supplemented modules are given. It is shown that (1) Arbitrary direct sum of -cofinitely supplemented modules is -cofinitely supplemented. (2) A ring R is semiperfect if and only if every free R-module is -cofinitely supplemented. In addition, if M has the summand sum property, then M is -cofinitely supplemented iff every maximal submodule has a supplement that is a direct summand of M.     

On τ-Extending Modules

Motivated by [2] and [6], we introduce a generalization of extending (CS) modules by using the concept of τ-large submodule which was defined in [9]. We give some properties of this class of modules and study their relationship with the familiar concepts of τ-closed, τ-complement submodules and the other generalization of extending modules (τ-complemented, τ-CS, s−τ-CS modules). We are also interested in determining when a τ-divisible module is τ-extending. For a τ-extending module M with C3, we obtain a decomposition theorem that there is a submodule K of M such that M = t(M) ? KM = \tau (M)\,\oplus\,K and K is τ (M)-injective. We also treat when a direct sum of τ-extending modules is τ-extending.     

On Graded Distributive Modules

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Σ-Extending Modules, Σ-Lifting Modules,and Proper Classes

We introduce generalizations of extending modules and lifting modules relative to proper classes of short exact sequences of modules, and we characterize modules for which any direct sums of copies of them are such relative extending modules or relative lifting modules.     

Normal Forms of Modules over Admissible Algebras with Formal Two-ray Modules

The aim of the paper is to classify the indecomposable modules and describe the Auslander-Reiten sequences for the admissible algebras with formal two-ray modules.