分次Morita对偶，Morita对偶与Smash积

设Ｃ和ｒ都是群，是Ｇ－型分次环，是Γ－型分次环．是双分次模，Ｒ＃Ｇ是Ｒ的Ｓｍａｓｈ积，Ａ＃Γ是Ａ的Ｓｍａｓｈ积。令Ｗ＝（＿ｇＵ＿（σ－１））＿（ｇ，σ）即（ｇ，σ）位置取＿ｇＵ＿（σ－１）的元素的｜Ｇ｜×｜Γ｜矩阵的全体组成的集合，且每个矩阵的每行和每列的非零元只有有限个，按矩阵运算，Ｗ构成（Ｒ＃６，Ａ＃Γ）双模。则＿ＲＵ＿Ａ定义了一个分次Ｍｏｒｉｔａ对偶当且仅当＿（Ｒ＃Ｇ）Ｗ＿（Ａ＃Γ）定义了一个Ｍｏｒｉｔａ对偶。     

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．     

G-集分次模与Morita Context

对任意群Ｇ， Ｈ≤Ｇ，［１］研究了Ｇ－分次环Ｒ与有限可迁Ｇ－集的ｓｍａｓｈ积.在本文中我们对任意可迁Ｇ－集，讨论了一个关于Ｒ（Ｈ）与ｓｍａｓｈ积Ｒ＃Ｇ／Ｈ的Ｍｏｒｉｔａ ｃｏｎｔｅｘｔ，从而推广了［２］，［３］，［４］给出的关于Ｇ－分次环及其与群Ｇ的ｓｍａｓｈ积的一些重要结果．     

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．     

环的优越扩张与凝聚维数

设Ｓ和Ｒ是环．本文证明了若下述条件之一成立，则Ｓ和Ｒ具有相同的凝聚维数：（１）Ｓ是Ｒ的优越扩张；（２）Ｓ和ＭＭｏｒｉｔａ等价．作为上述结果的推论，我们证明了环Ｒ和下述环类具有相同的凝聚维数：（ｉ）Ｒ上的矩阵环Ｍｎ（Ｒ）；（ｉ）Ｒ和有限群Ｇ（要求｜Ｇ｜－１∈Ｒ）的斜群环；（ｉｉ）Ｓｍａｓｈ积Ｒ＃Ｇ＊（要求Ｇ是有限群且｜Ｇ｜－１∈Ｒ，Ｒ是Ｇ分次环）     

偏序集上的Mobius反演公式（续）

偏序集上的Ｍｏｂｉｕｓ反演公式（续）万哲先（中国科学院）５局部有限偏序集上的关联代数和推广的Ｍｏｂｉｕｓ反演公式局部有限偏序集Ｐ上的Ｍｏｂｉｕｓ函数可以看作是Ｐ上的关联代数中的元素，现在先给出关联代数的定义．定义７设Ｐ是局部有限偏序集，Ｒ是有单位元１...     

投射自由环的Morita特征

本文研究投射自由环的Ｍｏｒｉｔａ特征,证明了投射自由环具有Ｍｏｒｉｔａ不变性．     

偏序集的关联环的同构问题

设Ｘ是局部有限偏序集（或拟序集），Ｒ是含１的结合环，Ⅰ（Ｘ，Ｒ）是Ｒ上Ｘ的关联环，关联环的同构问题是指：问题１：怎样的环，能使环同构Ⅰ（Ｘ，Ｒ）Ⅰ（Ｘ，Ｒ）推出偏序集之间的同构Ｘ芒Ｘ’？问题２：怎样的环或偏序集，能使环同构Ⅰ（Ｘ，Ｒ）Ⅰ（Ｘ，Ｓ）推出Ｒ　Ｓ？本文证明了对唯一幂等元环（非交换），问题１有正面回答；对问题２，我们证明了对交换不可分解环Ｒ、Ｓ，由环同构Ⅰ（Ｘ，Ｒ）Ⅰ（Ｘ，Ｒ）可得到Ｒ＝Ｓ，Ｘ＝Ｘ’。     

泛剩余交

本文建立了泛剩余交理论，并揭示了它与剩余交和一般剩余交的关系，得到：在交换局部环Ｒ的扩张Ｓ＝Ｒ（Ｘ）＝Ｒ［Ｘ］_{mＲ［Ｘ］}中，存在ＩＳ的一个ｓ－剩余交ＵＲＩ（ｓ；Ｉ），使得对Ｉ在Ｒ中的任意ｓ－剩余交Ｊ，ＵＲＩ（ｓ；Ｉ）是Ｊ的本质形变，且是Ｉ的一般ｓ－剩余交ＲＩ（ｓ；Ｉ）局部化ＲＩ（ｓ；Ｉ）_{mＲ［Ｘ］}．并给出了一些应用，为研究剩余交和一般剩余交提供了工具．     

特征模对环的刻划

设Ｒ是一个环，Ｍ是一个左Ｒ摸，Ｍ＊＝ＨｏｍＺ（Ｍ，Ｑ／Ｚ）为Ｍ的特征模．Ｒ．Ｒ．Ｃｏｌｂｙ和Ｔ．Ｊ．Ｃｈｏａｔｈａｎ等人利用特征模对ＩＦ环、凝聚环、Ｎｏｅｔｈｅｒ环、Ａｒｔｉｎ环作出了一些非常好的刻划．本文利用特征模对更为广泛的一些环作出了较有意义的刻划．     

广义幂级数环的Morita对偶

设A,B是有单位元的环, (S,≤)是有限生成的Artin的严格全序幺半群, AMB是双模.本文证明了双模[[AS,≤]][MS,≤][[BS,≤]]定义一个Morita对偶当且仅当 AMB定义一个Morita对偶且A是左noether的,B是右noether的.因此A上的广 义幂级数环[[AS,≤]]具有Morita对偶当且仅当A是左noether的且具有由双模AMB 诱导的Morita对偶,使得B是右noether的.     

半群分次环上的Morita对偶

本文主要研究了半群分次环上的Morita对偶问题,讨论了半群分次模范畴上满足某种条件的对偶函子与双分次双模之间的等价关系.得到重要定理：半群双分次$R$-$A$双模$Q$定义一个半群分次Morita对偶当且仅当${}_RQ_A$是分次忠实平衡的,且${\rm Ref}({}_RQ)$, ${\rm Ref}(Q_A)$对分次子模和分次商模是封闭的.     

Morita duality for monoids

In this paper Morita duality for monoids is introduced. Necessary and sufficient conditions for two monoids S and T to be Morita dual are given. Moreover, it is shown that if S and T are Morita dual monoids, then S and U are Morita dual if and only if T and U are Morita equivalent. In addition, every finite monoid having Morita duality is selfdual and even reflexive.     

The strong persistence property and symbolic strong persistence property

Czechoslovak Mathematical Journal - Let I be an ideal in a commutative Noetherian ring R. Then the ideal I has the strong persistence property if and only if (Ik+1: RI) = Ik for all k, and I has...     

Duality in algebra and topology

We apply ideas from commutative algebra, and Morita theory to algebraic topology using ring spectra. This allows us to prove new duality results in algebra and topology, and to view (1) Poincaré duality for manifolds, (2) Gorenstein duality for commutative rings, (3) Benson–Carlson duality for cohomology rings of finite groups, (4) Poincaré duality for groups and (5) Gross–Hopkins duality in chromatic stable homotopy theory as examples of a single phenomenon.     

Gr-Morita对偶与Morita对偶

设G为有单位元e的群R=(?)R_x和A=(?)A_x都是有单位元的G型强分次环,U=(?)U_z是分次(R,A)一双模.本文主要证明了_RU_A导出一个Gr—Morita对偶当且仅当R_eU_(eAc)导出一个Morita对偶.     

Morita Duality for the Rings of Generalized Power Series

Let A, B be associative rings with identity, and (S, ≤) a strictly totally ordered monoid which is also artinian and finitely generated. For any bimodule _A M _B , we show that the bimodule _{[[ AS,≤ ]]}[M ^{S ,≤}]_{[[ BS, ≤ ]]} defines a Morita duality if and only if _A M _B defines a Morita duality and A is left noetherian, B is right noetherian. As a corollary, it is shown that the ring [[A ^{S ,≤}]] of generalized power series over A has a Morita duality if and only if A is a left noetherian ring with a Morita duality induced by a bimodule _A M _B such that B is right noetherian. Received April 13, 1999, Accepted December 12, 1999     

Poset Loops

Given a ring and a locally finite poset, an incidence loop or poset loop is obtained from a new and natural extended convolution product on the set of functions mapping intervals of the poset to elements of the ring. The paper investigates the interplay between properties of the ring, the poset, and the loop. The annihilation structure of the ring and extremal elements of the poset determine commutative and associative properties for loop elements. Nilpotence of the ring and height restrictions on the poset force the loop to become associative, or even commutative. Constraints on the appearance of nilpotent groups of class 2 as poset loops are given. The main result shows that the incidence loop of a poset of finite height is nilpotent, of nilpotence class bounded in terms of the height of the poset.     

Weak Morita Duality

Abstract

The notion of a weak Morita duality between subcategories of Mod-R and S-Mod is introduced and investigated. This concept includes dualities induced by cotilting modules over finite dimensional algebras, the R-dual for both noetherian rings of injective dimension ≤ 1 and the Matlis duality domains, as well as the first author's generalized Morita duality. Moreover it yields a “cotilting theorem” dual to the classical tilting theorem, and a characterization of generalized Morita duality.