A proof that commutative artinian rings are noetherian

Abstract

Integrals in Hopf algebras are an essential tool in studying finite dimensional Hopf algebras and their action on rings. Over fields it has been shown by Sweedler that the existence of integrals in a Hopf algebra is equivalent to the Hopf algebra being finite dimensional. In this paper we examine how much of this is true Hopf algebras over rings. We show that over any commutative ring R that is not a field there exists a Hopf algebra H over R containing a non-zero integral but not being finitely generated as R-module. On the contrary we show that Sweedler's equivalence is still valid for free Hopf algebras or projective Hopf algebras over integral domains. Analogously for a left H-module algebra A we study the influence of non-zero left A#H-linear maps from A to A#H on H being finitely generated as R-module. Examples and application to separability are given.     

相关Hopf模的对偶

本文的目的就是给出相关Hopf模的对偶性质.在第一部分,证明了相关Hopf模的对偶模仍是相关Hopf模.特别地,Hopf模的对偶仍是Hopf模.在第二、第三两部分,分别给出相关Hopf模的对偶相关Hopf模的基本结构定理及Maschke定理.     

Modules Characterized by Injectivity Clases

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模代数的形变理论

本文构作了与模代数HA相适应的“形变复形”CH*(H,A),并利用形变复形的上同调群来刻画模代数HA的形变,证明了HA的无穷小形变的等价类是与形变复形的2阶上同调群HH2(H,A)是一一对应的,且当HH2(H,A)=0时,模代数HA为刚性的.     

基于改进型群层次分析法的装备维修过程模块质量评估方法研究

深入研究了群AHP理论(AHP, Analytic Hierarchy Process,层次分析法),运用距离标度对其进行了改进。并分析、界定了装备维修过程模块,建立了质量评估指标体系,运用了距离改进群AHP方法评估装备维修过程模块质量,为通过计算机进行装备维修质量控制提供了方法。     

G-M制冷机热力计算模块设计

主要介绍G-M制冷系统热力性质参数及循环性能参数的计算模块的设计,该模块用Visual Basic 6.0语言进行开发,具有良好的人机交互界面,使用方便,运行可靠.将该模块用于常规G-M制冷机的热力计算,结果合理,达到了预期的要求.因此,该模块在小型低温制冷机结构设计方面具有广泛的推广价值和应用前景.     

Centre of an algebra

Motivated by algebraic structures appearing in Rational Conformal Field Theory we study a construction associating to an algebra in a monoidal category a commutative algebra (full centre) in the monoidal centre of the monoidal category. We establish Morita invariance of this construction by extending it to module categories.As an example we treat the case of group-theoretical categories.     

Operator decomposition of graphs and the reconstruction conjecture

We present the method of proving the reconstructibility of graph classes based on the new type of decomposition of graphs — the operator decomposition. The properties of this decomposition are described. Using this decomposition we prove the following. Let P and Q be two hereditary graph classes such that P is closed with respect to the operation of join and Q is closed with respect to the operation of disjoint union. Let M be a module of graph G with associated partition (A,B,M), where A∼M and B⁄∼M, such that G[A]∈P, G[B]∈Q and G[M] is not (P,Q)-split. Then the graph G is reconstructible.