Comatrix Corings and Galois Comodules over Firm Rings

We construct comatrix corings on bimodules without finiteness conditions by using firm rings. This leads to the formulion of a notion of Galois coring which plays a key role in the statement of a Noncommutative Faithfully Flat Descent for comodules which generalizes previous versions. In particular, infinite comatrix corings fit in our general theory. Presented by A. Verschoren.     

Fundamental Constructions for Coalgebras, Corings, and Comodules

We study the various categories of corings, coalgebras, and comodules from a categorical perspective. Emphasis is given to the question which properties of these categories can be seen as instances of general categorical resp. algebraic results. However, we also obtain new results concerning the existence of limits and of factorizations of morphisms.      

Group Corings

We introduce group corings, and study functors between categories of comodules over group corings, and the relationship to graded modules over graded rings. Galois group corings are defined, and a Structure Theorem for the G-comodules over a Galois group coring is given. We study (graded) Morita contexts associated to a group coring. Our theory is applied to group corings associated to a comodule algebra over a Hopf group coalgebra. This research was supported by the research project G.0622.06 “Deformation quantization methods for algebras and categories with applications to quantum mechanics” from Fonds Wetenschappelijk Onderzoek-Vlaanderen. The third author was partially supported by the SRF (20060286006) and the FNS (10571026).     

Fully Coprime Comodules and Fully Coprime Corings

Prime objects were defined as generalization of simple objects in the categories of rings (modules). In this paper we introduce and investigate what turns out to be a suitable generalization of simple corings (simple comodules), namely fully coprime corings (fully coprime comodules). Moreover, we consider several primeness notions in the category of comodules of a given coring and investigate their relations with the fully coprimeness and the simplicity of these comodules. These notions are applied then to study primeness and coprimeness properties of a given coring, considered as an object in its category of right (left) comodules. Supported by King Fahd University of Petroleum and Minerals, Research Project # INT/296.     

Rings of the Morita Contexts and Semihereditary Maximal V-Orders

Some necessary and sufficient conditions are given for rings of the Morita contexts to be semihereditary maximal V-orders, Prüfer rings and Dubrovin valuation rings.     

正则QB环上的Morita Contexts

假设T是Morita Contexts(A,B,M,N,φ,φ)的环,如果A和B是正则QB环,本文证明了T中任何元素都是幂等元和伪逆元之和.     

Morita Equivalence for Rings with Involution

We develop the theory of Morita equivalence for rings with involution, and we show the corresponding fundamental representation theorem. In order to allow applications to operator algebras, we work within the class of idempotent nondegenerate rings. We also prove that two commutative rings with involution are Morita *-equivalent if and only if they are *-isomorphic.     

Morita contexts and Galois theory for weak Hopf comodulelike algebras

We investigate the Morita context and graded cases for weak group corings and derive some equivalent conditions for μ to be surjective. Furthermore, we develop Galois theory for weak group corings. As an application, we give Galois theory for comodulelike algebras over a weak Hopf group coalgebra.     

Study of Morita contexts

Matrix congruence can be used to mimic linear maps between homogeneous quadratic polynomials in n variables. We introduce a generalization, called standard-form congruence, which mimics affine maps between non-homogeneous quadratic polynomials. Canonical forms under standard-form congruence for three-by-three matrices are derived. This is then used to give a classification of algebras defined by two generators and one degree two relation. We also apply standard-form congruence to classify homogenizations of these algebras.     

群分次环和Morita等价

设Ｒ是Ｇ－分次环，Ｈ是Ｇ的子群，本文通过构造两个Ｍｏｒｉｔａｃｏｎｔｅｘｔ，给出了环Ｂ｛Ｈ｝Ｍｏｒｉｔａ等价于Ｒ＿Ｈ，Ｒ＃Ｇ￣＊Ｍｏｒｉｔａ等价于Ｒ＿Ｈ＃Ｈ￣＊的充分必要条件．     

The Morita-Takeuchi Theory for Quotient Categories

Abstract

We study the Morita-Takeuchi context connecting two coalgebras which is dual to the Morita context for algebras. We show that every Morita-Takeuchi context, connecting two coalgebras C and D, leads to an equivalence between quotient categories of the comodule categories ^C M and ^D M. Afterwards we introduce a special Morita-Takeuchi context, called closed, and show that there is a bijection between isomorphism types of closed contexts and isomorphism types of category equivalences between quotient categories of ^C M and ^D M determined by localizing subcategories. This represents a dualization of the classical Morita theorems. Finally we show that from every general context one can construct a closed one.     

Rational Modules for Corings

Abstract

The so called dense pairings were studied mainly by Radford in his work on coreflexive coalegbras over fields. They were generalized in a joint paper with Gómez-Torricillas and Lobillo to the so called rational pairings over a commutative ground ring R to study the interplay between the comodules of an R-coalgebra C and the modules of an R-algebra A that admits an R-algebra morphism κ : A → C*. Such pairings, satisfying the so called α-condition, were called in the author's dissertation measuring α-pairings and can be considered as the corner stone in his study of duality theorems for Hopf algebras over commutative rings. In this paper we lay the basis of the theory of rational modules of corings extending results on rational modules for coalgebras to the case of arbitrary ground rings. We apply these results mainly to categories of entwined modules (e.g., Doi-Koppinen modules, alternative Doi-Koppinen modules) generalizing results of Doi, Koppinen, Menini et al.     

Descent Theory and Morita Theory for Ultrametric Banach Modules

In this paper we consider ultrametric Banach modules over commutative ultrametric Banach algebras with unit. We study the descent problem along a morphism f: R S of such algebras and show that descent morphisms coincide with weak retracts. We give further conditions for having an effective descent morphism or for having a Morita equivalence between the corresponding categories of ultrametric Banach modules.     

A Zariski Topology for Bicomodules and Corings

In this paper we introduce and investigate top (bi)comodules of corings, that can be considered as dual to top (bi)modules of rings. The fully coprime spectra of such (bi)comodules attains a Zariski topology, defined in a way dual to that of defining the Zariski topology on the prime spectra of (commutative) rings. We restrict our attention in this paper to duo (bi)comodules (satisfying suitable conditions) and study the interplay between the coalgebraic properties of such (bi)comodules and the introduced Zariski topology. In particular, we apply our results to introduce a Zariski topology on the fully coprime spectrum of a given non-zero coring considered canonically as duo object in its category of bicomodules. Supported by King Fahd University of Petroleum & Minerals, Research Project # INT/296.     

Hermitian Morita Theory and Unitary Groups of Modules over Semisimple Artinian Rings

This article investigates isomorphisms between certain subgroups of the projective unitary groups of hermitian modules over semisimple Artinian rings with anti-structures. These subgroups contain the commutator subgroups of the projective unitary groups. Specifically, the article provides conditions under which these isomorphisms are induced by and the underlying rings are connected by hermitian Morita equivalences (HMEs). This article introduces the hyperbolic length of a module as well as the concept of generalized hyperbolic modules over simple Artinian rings, and over semisimple Artinian rings with anti-structures. The article shows that the stated isomorphisms are induced by HMEs if all the following conditions hold: (a) the hermitian forms are nonsingular and trace valued; (b) the modules in question are generalized hyperbolic; (c) the hyperbolic length equals three or is greater than or equal to five (hyperbolic length is greater than or equal to five in the semisimple case). Significantly, the condition of the hyperbolic length of a module greater than or equal to m is satisfied by a set of modules larger than or equal to those satisfying the condition of the Witt index of the module greater than or equal to m.     

有足够幂等元的群分次环上的Morita对偶

首先定义了具有足够幂等元的群分次环的Ｓｍ ａｓｈ 积并讨论其性质,其次,借助Ｓｍ ａｓｈ 积给出了这类分环上的分次Ｍｏｒｉｔａ对偶的刻划．     

可分解半群的Morita等价

设S,R是可分解半群.记US-FAct={sM∈S-Act|SM=M且SHoms(S,M)≌M],给出了范畴US-FAct与UR-FAct等价的刻划;S分别强Morita等价于一个夹层半群、局部单位半群、幺半群和群的条件;S是完全单半群当且仅当S强Morita等价于一个群且对任何指标集I,S SHoms(S,i∈I S)→i∈I S,s t·f→(st)f,是同构.     

Morita Invariance in Cyclic Homology for Nonunital Algebras

Let E and F be two linear subspaces of an algebra A over a field of characteristic zero and let A(EF), A(FE) be the subalgebras of A generated by EF and FE, respectively. We show that A(EF) and A(FE) have the same periodic cyclic homology.     

Cyclic Codes Over\mathbb{Z}_{4} of Even Length

We determine the structure of cyclic codes over for arbitrary even length giving the generator polynomial for these codes. We determine the number of cyclic codes for a given length. We describe the duals of the cyclic codes, describe the form of cyclic codes that are self-dual and give the number of these codes. We end by examining specific cases of cyclic codes, giving all cyclic self-dual codes of length less than or equal to 14. San Ling - The research of the second named author is partially supported by research Grants MOE-ARF R-146-000-029-112 and DSTA R-394-000-011-422.     

Morita Context and Morita—Like Equivalence for the xst—Rings

The notion of the xst-rings was introduced by Garcfa and Marfn [5] in 1999.In this paper,we cousider Morita context,Morita-like equivalence and the exchange property for the xst-rings.The results of the first Morita theorem are generalized to the xst-rings.So we obtain an important Morita-like equivalence of the xst-rings,from which,as an immediate consequence,we deduce the main result of Xu-Shum-Turner [4] and the standard Morita equivalence,A-Mn(A),for a unital ring A.Moreover,we describe the properties of those well-known intermediate matrix rings,and show that the exchange property for a unital ring A coincides with the one for any Mn(A) as well as any intermediate matrix ring sitting between FM1(A) and FCг(A),which is an extension of a well-known result obtained by Nicholson[7].