Varieties of two-dimensional cylindric algebras.¶Part I: Diagonal-free case

We investigate the lattice of all subvarieties of the variety Df ₂ of two-dimensional diagonal-free cylindric algebras. We prove that a Df ₂-algebra is finitely representable if it is finitely approximable, characterize finite projective Df ₂-algebras, and show that there are no non-trivial injectives and absolute retracts in Df ₂. We prove that every proper subvariety of Df ₂ is locally finite, and hence Df ₂ is hereditarily finitely approximable. We describe all six critical varieties in , which leads to a characterization of finitely generated subvarieties of Df ₂. Finally, we describe all square representable and rectangularly representable subvarieties of Df ₂. Received May 25, 2000; accepted in final form November 2, 2001.     

On Ding Projective Complexes

In the paper, Ding projective modules and Ding projective complexes are considered. In particular, it is proven that Ding projective complexes are precisely the complexes X for which each X_m is a Ding projective R-module for all m ∈ Z.     

Some remarks on projective generators and injective cogenerators

In this paper,we give a relationship between projective generators(resp.,injective cogenerators) in the category of R-modules and the counterparts in the category of complexes of R-modules.As a consequence,we get that complexes of W-Gorenstein modules are actually W-Gorenstein complexes whenever W is a subcategory of R-modules satisfying W⊥W,where W is the subcategory of exact complexes with all cycles in W.We also study when all cycles of a W-Gorenstein complexes are W-Gorenstein modules.     

Fibonacci代数的Hochschild同调群

设Λd是Fibonacci代数,基于对Bardzell极小投射双模分解的细致分析,用组合的方法清晰地计算了Fibonacci代数Λd的各阶Hochschild同调群的维数.     

量子代数的内射模与Tilting模

A=Z[ν] m ' m是 Z[ν]的由ν- 1和奇素数 p生成的理想 .U是 A上的量子代数 .设 k是特征为零的代数闭域 .A→ K (ν|→ξ)是代数同态 ,并假定ξ不是 1的根或ξ是 p次本原根 .命Uk=U k A.J是 UK- Tilting模范畴 .对 λ∈ X+,M(λ)表首权为 λ的不可分解 UK- Tilting模 .本文证明了 ,对每个λ∈ X+,M(λ)作为 Uk 模是内射的当且仅当λ- (p- 1 )ρ∈ X+.我们还给出了内射 Uk模的若干充要条件 .     

投射根和亚投射模

本文研究了投射根和亚投射模的性质.利用Noether环、Noether模的性质给出了亚投射模、投射根是投射的条件,环的投射根为零、幂等的条件.最后,给出了一些例子.本文的大部分结果是陈焕艮等人的一系列主要结果的推广.     

Gorenstein Injective Envelopes of Artinian Modules

Let R be a commutative Noetherian ring and A an Artinian R-module. We prove that if A has finite Gorenstein injective dimension, then A possesses a Gorenstein injective envelope which is special and Artinian. This, in particular, yields that over a Gorenstein ring any Artinian module possesses a Gorenstein injective envelope which is special and Artinian.     

二元广义外代数的Hochschild同调群

设Aq=k／(x2,xy+qyx,y2)是含有两个变量的广义外代数,基于Buch- weitz等人构造的极小投射双模解,广义外代数的各阶Hochschild同调群的维数被清晰地计算．     

长度有限的模的对偶

本文推广了局部对偶理论并讨论了长度为有限的模的同调维数．     

弱直投射模和弱直内射模

引入了弱直投射和弱直内射模的概念，给出了它们的一些性质。使用弱直投射和弱直内射模刻划了遗传环、半遗传环、半单环和ＱＦ－环。     

On FC-Purity and I-Purity of Modules and Köthe Rings

In this article, several characterizations of certain classes of rings via FC-purity and I-purity are considered. Among others results, it is shown that every I-pure injective left R-module is projective if and only if every FC-pure projective left R-module is injective, if and only if, R is a semisimple ring. In particular, the structures of FC-pure projective and I-pure projective modules over a left Artinian ring are completely described. Also, it is shown that every left R-module is FC-pure projective if and only if every indecomposable left R-module is a finitely presented cyclic R-module, if and only if, R is a left Köthe ring. Finally, we introduce FC-pure flatness and I-pure flatness of modules and several characterizations of these notions are given. In particular, we show that a commutative ring R is quasi-Frobenius if and only if R is an Artinian ring and I-pure injective, if and only if, R is an Artinian ring and the injective envelope E(R) is an FC-pure projective R-module.     

Gorenstein Modules and Dualizing Modules

In this article, we study the characterizations of Gorenstein injective left S-modules and finitely generated Gorenstein projective left R-modules when there is a dualizing S-R-bimodule associated with a right noetherian ring R and a left noetherian ring S.     

On maximal injective subalgebras of tensor products of von Neumann algebras

Let Mi be a von Neumann algebra, and Bi be a maximal injective von Neumann subalgebra of Mi, i=1,2. If M₁ has separable predual and the center of B₁ is atomic, e.g., B₁ is a factor, then is a maximal injective von Neumann subalgebra of . This partly answers a question of Popa.     

Gorenstein injective modules and local cohomology

In this paper we assume that is a Gorenstein Noetherian ring. We show that if is also a local ring with Krull dimension that is less than or equal to 2, then for any nonzero ideal of , is Gorenstein injective. We establish a relation between Gorenstein injective modules and local cohomology. In fact, we will show that if is a Gorenstein ring, then for any -module its local cohomology modules can be calculated by means of a resolution of by Gorenstein injective modules. Also we prove that if is -Gorenstein, is a Gorenstein injective and is a nonzero ideal of , then is Gorenstein injective.

     

Commutative group algebras of -summable abelian groups

In this note we study the commutative modular and semisimple group rings of -summable abelian -groups, which group class was introduced by R. Linton and Ch. Megibben. It is proved that is -summable if and only if is -summable, provided is an abelian group and is a commutative ring with 1 of prime characteristic , having a trivial nilradical. If is a -summable -group and the group algebras and over a field of characteristic are -isomorphic, then is a -summable -group, too. In particular provided is totally projective of a countable length.

Moreover, when is a first kind field with respect to and is -torsion, is -summable if and only if is a direct sum of cyclic groups.

     

Central Pairs, Galois Theory and Automorphic Forms

A central pair over a field k of characteristic 0 consists of a finite Abelian group which is equipped with a central 2-cocycle with values in the multiplicative group k ^* of k. In this paper we use specific central pairs to construct a class of projective representations of the absolute Galois group G _k of k and if k is a number field we investigate the liftings of these projective representations to linear representations of G _k . In particular we relate these linear representations to automorphic representations. It turns out that some of these automorphic representations correspond to certain indefinite modular forms already constructed by E. Hecke.     

ON THE CENTRAL LIMIT THEOREM IN PRODUCT SPACES

ＯＮＴＨＥＣＥＮＴＲＡＬＬＩＭＩＴＴＨＥＯＲＥＭＩＮＰＲＯＤＵＣＴＳＰＡＣＥＳＳＵＺＨＯＮＧＧＥＮＡｂｓｔｒａｃｔ：ＳｕｐｐｏｓｅｔｈａｔＥａｎｄＦａｒｅｓｅｐａｒａｂｌｅＢａｎａｃｈｓｐａｃｅｓ，ＸａｎｄＹａｒｅｉｎｄｅｐｅｎｄｅｎｔｓｙｍｍｅｔｒ...     

正则局部环的判别法（英）

本文给出了非Artin的Noether局部环(A, m)为正则环的充要条件.设n为正整数，则(A, m)为正则环的充要条件为A/mⁿ的投射维数或mⁿ的内射维数是有限的.     

关于扭Hopf代数的一些注记

In this paper,we get some properties of the antipode of a twisted Hopf algebra.We proved that the graded global dimension of a twisted Hopf algebra coincides with the graded projective dimension of its trivial module k,which is also equal to the projective dimension of k.