Hereditary and Formally Smooth Coalgebras

We define formally smooth coalgebras and we study their relation with hereditary coalgebras. The main result of the paper establishes that a coalgebra with separable coradical is hereditary if and only if it is formally smooth if and only if it is the cotensor coalgebra , where C₀ is the coradical of C and N is a certain (C₀,C₀)–bicomodule. Presented by A. Verschoren Mathematics Subject Classifications (2000) primary: 16W30; secondary: 18G20. D. Ştefan: This author thanks the members of the Algebra Department of the University of Granada for their warm hospitality. He is especially grateful to Pascual Jara for the kind invitation to visit University of Granada. This research was partially supported by CERES, Contract 4-147.     

环上的σ-导子

本文研究了σ－导子的扩张问题，并且在本原环上刻化了s－导子。     

三角环上的σ-双导子

本文利用极大右商环证明了一类三角环上的σ-双导子可以表示成极值σ-双导子和内σ-双导子之和.     

集合范畴上超滤函子的余代数

定义了集合范畴上的超滤函子F_u(-),并研究了相关性质.包括函子F_u(-)在有限集上保拉回,一个集合的子集成为F_u-子余代数的充要条件,以及两个余代数之间的态射是F_u-余代数同态的充要条件,子集成为子余代数的充要条件,最后以拓扑空间作为F_u-余代数的具体实例,研究了拓扑空间的连续映射与超滤函子的余代数同态之间的关系.     

Associated Graded Algebras and Coalgebras

We investigate the notion of associated graded coalgebra (algebra) of a bialgebra with respect to a subbialgebra (quotient bialgebra) and characterize those which are bialgebras of type one in the framework of abelian braided monoidal categories.     

On modules which force homogeneous maps to be linear

Let be a ring with identity. We characterize in terms of the left ideal structure of when every homogeneous map between nonsingular -modules is linear and answer some earlier questions of the author that remained open.

     

广义路余代数和广义对偶Gabriel定理

通过将箭图的每个顶点放置一个k-余代数,首先引进了广义路余代数的概念,其次给出了广义路余代数的一些基本性质,还讨论了同构问题.证明了两个正规广义路余代数是同构的当且仅当他们的箭图及对应顶点上的单余代数是同构的.对于满足Codim C_0■1余代数C,证明了对偶Wedderburn-Malcev定理成立.作为广义路余代数的一个应用,推广了点余代数的对偶Gabriel定理.     

Maximal Subrings and Covering Numbers of Finite Semisimple Rings

We classify the maximal subrings of the ring of n×n matrices over a finite field, and show that these subrings may be divided into three types. We also describe all of the maximal subrings of a finite semisimple ring, and categorize them into two classes. As an application of these results, we calculate the covering number of a finite semisimple ring.     

关于拟三角Hopf代数的Cylinder余代数和Cylinder余积(英)

本文引入两个概念,即,关于拟三角双代数的cylinder余代数和cylinder余积,并指出存在一个反余代数同构：(H,■)≌(H,■),其中(H,■)是cylinder余积,(H,■)是辫余积,对任意有限维Hopf代数H,我们证明Drinfel'd量子偶(D(H),■_(D(H)))是cylinder余积．设(H,H,R)是余配对Hopf代数,如果R∈Z(H■H),则通过两次扭曲,我们可以构造扭曲余代数(H~■)R~(-1),它的余乘法恰是cylinder余积．而且对任意的广义Long重模,通过cylinder扭曲,我们可以构造Yang-Baxter方程,四辫对和Long方程．     

广义内射性与von Neumann正则性

引入ＩＣＥ－内射它是拟内射的推广，利用它来研究ｖｏｎ Ｎｅｕｍａｎｎ正则环、遗传环、Ａｒｔｉｎ半单环。     

关于拟P-内射模的一些注记(英文)

设 R是一个环 .一个右 R-模 M叫做拟 P-内射的 ,如果 M的每个 M-循环子模到 M的任一个 R-同态都能扩展到 M.假设 M是一个自生成子的拟 P-内射模 .在这篇文章中 ,我们表明如果这样一个模是一个 CF-模 (特别地 ,CS-模 ) ,那么 S/J(S)是正则的 ,其中 S=End(MR) .进一步 ,如果 S是半素环 ,那么 M的每个极大核是 M的一个直和项 .这些结果扩展了 P-内射环的一些结果     

Dual Coalgebras of Jordan Bialgebras and Superalgebras

W. Michaelis showed for Lie bialgebras that the dual coalgebra of a Lie algebra is a Lie bialgebra. In the present article we study an analogous question in the case of Jordan bialgebras. We prove that a simple infinite-dimensional Jordan superalgebra of vector type possesses a nonzero dual coalgebra. Thereby, we demonstrate that the hypothesis formulated by W. Michaelis for Lie coalgebras fails in the case of Jordan supercoalgebras.     

Quasi-cotilting modules and hereditary quasi-cotilting modules

In this paper, we firstly give some basic properties on quasi-cotilting modules. With the help of these properties, we obtain a Quasi-cotilting Theorem (see Theorem 2.8). In particular, the Cotilting Theorem in [5 Colpi, R. R. (2000). Cotilting bimodules and their dualities. Lect. Notes Pure Appl. Math. 210:81–94. [Google Scholar]] is a corollary of our result. Finally, we introduce a new notion, hereditary quasi-cotilting modules, and mainly prove that the class consisting of all Δ-reflexive modules for a hereditary quasi-cotilting module is closed under submodules.     

Irreducibility in the Total Ring of Quotients

Let R be a ring whose total ring of quotients Q is von Neumann regular. We investigate the structure of R when it admits an ideal that is irreducible as a submodule of the total ring of quotients. We characterize those rings which contain a maximal ideal that is irreducible in its total ring of quotients Q. An integral domain has a Q-irreducible ideal which is a maximal ideal if and only if R is a valuation domain. We show that when the total ring of quotients of R is von Neumann regular, then having a maximal ideal that is Q-irreducible is equivalently to certain valuation like properties, including the property that the regular ideals are linearly ordered.     

Preserving Zeros of xy and xy*

We describe surjective additive maps θ: A → B which preserve zero products, where A is a ring with a nontrivial idempotent and B is a prime ring. We also characterize surjective additive maps θ: A → B such that for all x, y ∈ A we have θ(x)θ(y)* = 0 if and only if xy* = 0. Here A is a unital prime ring with involution that contains a nontrivial idempotent and B is a prime ring with involution.     

A Connection Between the Reticulation of a Ring of Quotients and the Localization Lattice of the Reticulation of a Commutative Ring

Through this article, R denotes a commutative ring with identity. The aim of this article is to construct a morphism of lattices between the reticulation of a ring of quotients (of a commutative ring with respect to a Gabriel topology) and the localization lattice of the reticulation of the initial ring. The article is structured as follows: The first section introduces all the necessary notions required to simplify the reading of the article. The second section presents some properties of the topology induced by a Gabriel topology on the reticulation L(R). In the third section the morphism δ, which achieves the intended link, is constructed. The last section studies the defined morphism in some particular cases.