A variant of the Reynolds operator

Let be a linearly reductive group over a field , and let be a -algebra with a rational action of . Given rational --modules and , we define for the induced -action on Hom a generalized Reynolds operator, which exists even if the action on Hom is not rational. Given an -module homomorphism , it produces, in a natural way, an -module homomorphism which is -equivariant. We use this generalized Reynolds operator to study properties of rational - modules. In particular, we prove that if is invariantly generated (i.e. ), then is a projective (resp. flat) -module provided that is a projective (resp. flat) -module. We also give a criterion whether an -projective (or -flat) rational --module is extended from an -module.

     

Essential Ideals in Rings of Measurable Functions

In this article, we study essential ideals, socle, and some related ideals of rings of real valued measurable functions. We also study the Goldie dimension of rings of measurable functions.     

On exchange rings with primitive factor rings artinian

Abstract

In this paper we introduce generalized ideal-stable regular rings. It is shown that if a regular ring R is a generalized I-stable ring, then every square matrix over I is the product of an idempotent matrix and an generalized invertible matrix and admits a diagonal reduction by some generalized invertible matrices.     

Wacker Oxidation of Methylenecyclobutanes: Scope and Selectivity in an Unusual Setting

Methylenecyclobutanes are found to undergo Wacker oxidation via a semi-pinacol-type rearrangement. Key to a successful process is the use of tert-butyl nitrite as oxidant, which not only enables efficient catalyst turn-over but also ensures high Markovnikov-selectivity under mild conditions. Thus, cyclopentanones (26 examples) can be accessed in an overall good yield and excellent selectivity (up to 97 % yield, generally >99 : 1 ketone:aldehyde ratio). Stereochemical analysis of the reaction sequence reveals migration aptitudes in line with related 1,2-shifts. By introducing a pyox ligand to palladium, prochiral methylenecyclobutanes can be desymmetrized, thus realizing the first enantioselective Wacker oxidation.     

On the notion of Cohen–Macaulayness for non-Noetherian rings

There exist many characterizations of Noetherian Cohen–Macaulay rings in the literature. These characterizations do not remain equivalent if we drop the Noetherian assumption. The aim of this paper is to provide some comparisons between some of these characterizations in non-Noetherian case. Toward solving a conjecture posed by Glaz, we give a generalization of the Hochster–Eagon result on Cohen–Macaulayness of invariant rings, in the context of non-Noetherian rings.     

Ideal-comparability over Regular Rings

We introduce the concept of ideal-comparability condition for regular rings. Let I be an ideal of a regular ring R. If R satisfies the I-comparability condition, then R is one-sided unit-regular if and only if so is R/I. Also, we show that a regular ring R satisfies the general comparability if and only if the following hold： （1） R/I satisfies the general comparability; （2） R satisfies the general I-comparability condition; （3） The natural map B（R） → B（R/I） is surjective.     

DYNAMICAL EVOLUTION OF URANIAN ε RING WITH MAPPING APPROACH

ＤＹＮＡＭＩＣＡＬＥＶＯＬＵＴＩＯＮＯＦＵＲＡＮＩＡＮεＲＩＮＧＷＩＴＨＭＡＰＰＩＮＧＡＰＰＲＯＡＣＨＺｈｏｕＪｉｌｉｎＳｕｎＹｉｓｕｉ（ＤｅｐａｒｔｍｅｎｔｏｆＡｓｔｒｏｎｏｍｙ，ＮａｎｊｉｎｇＵｎｉｖｅｒｓｉｔｙ，２１００９３，Ｎａｎｊｉｎｇ，...     

牛顿环测透镜曲率半径数据处理

在误差理论中,有多种处理数据的方法．文章详细介绍了用逐差法、最小二乘法、加权平均法处理牛顿环测透镜曲率半径的数据的方法和过程．     

Cyclization and Isomerization Reactionsin Silylhydrazine Chemistry

Summary.  This review describes the synthesis and isomerization reactions of cyclic silylhydrazines. Topics of discussion are the ring expansion of the three-membered Si(SiN₂) to the four-membered (SiN)₂ ring by lithiation of the (SiN₂) ring and by thermal silyl group insertion into the N*N bond, the ring expansion of a three-membered (SiN₂) to a five-membered (CSi₂N₂) ring by SiCH₂ insertion into the Si*N bond, the formation of isomeric four- and six-membered (SiN₂)₂ rings, the synthesis of five- and six-membered silylhydrazine rings, and the expansion of a five-membered (N₂Si₂N)N ring to the isomeric six-membered (SiNN)₂ ring. The mechanisms of the isomerizations are explained by quantum chemical calculations, and the results are verified by crystal structure determinations. Received February 23, 2001. Accepted (revised) April 24, 2001     

From chains to rings, and back again: structural rearrangements during evaporation and condensation

The mechanistic details of the sequence of structural changes that occur in weakly bound molecular complexes upon change in phase have yet to be presented in detail. Consider the structural rearrangements that occur at the interface between the liquid and its vapor for species such as hydrofluoric acid or the low molecular weight aliphatic acids. There are both experimental and theoretical evidence that liquid HF consists of extended zig-zag chains [1], whereas the vapor consists predominantly of non-planar cyclic structures [2]. Similarly, the cyclic structures of gaseous aliphatic acid dimers was established decades ago but in the solid and liquid phases, the molecular configurations are zig-zag chains [3].