有限置换模自同态环的可分性质

设Q是有限置换右R模,则EndR(Q)是可分环当且仅当对所有A,B∈FP(Q),A A≌A B≌B B A≤ B或B≤ A.作为应用得到了EndR(P Q)是可分环当且仅当EndRP和EndRQ为可分环,其中P,Q为有限置换右R模.     

有限置换模自同态环的可分性质

设Q是有限置换右R模,则End_R(Q)是可分环当且仅当对所有A,B∈FP(Q),A AA B B B A≤ B或 B≤A,作为应用得到了 End_R(P Q)是可分环当且仅当End_R P和End_R Q为可分环,其中P,Q为有限置换右R模。     

COMMUTATIVE RINGS WITH FINITE QUOTIENT FIELDS

Abstract

We consider the class of all commutative reduced rings for which there exists a finite subset T ? A such that all projections on quotients by prime ideals of A are surjective when restricted to T. A complete structure theorem is given for this class of rings,and it is studied its relation with other finiteness conditions on the quotients of a ring over its prime ideals.     

FINITE RINGS WITH COMMUTING NILPOTENT ELEMENTS

As a generalization of Wedderburn's theorem, Herstein [5] proved that a finite ring R is commutative, if all nilpotent elements are contained in the center of R. However a finite ring with commuting nilpotent elements is not necessarily commutative. Recently, in [9] and [10], Simons described the structure of finite rings R with J(R)² = 0 in a variety with definable principal congruences. In this paper, we will consider the difference between the finite commutative rings and the finite rings in which any two nilpotent elements commute with each other. As a consequence, we describe the structure of finite rings R with [J(R), J(R)] = 0 in a variety with definable principal congruences.     

TORSION-FREE ABELIAN GROUPS WITH FINITE RANK ENDOMORPHISM RINGS

Abstract

We show that if A and C are torsion-free abelian groups with ∩{ker f|f: A → C} = 0 = ∩{ker g|g: C → A}, and if A has a left Artinian quasi-endomorphism ring then A and C share a nonzero quasi-summand. Some consequences explored.     

分次环的有限正规扩张

本文研究了群分次环的有限正规分次扩张问题．利用经典环论方法，得到一个群分次环与其有限正规分次扩张环之间关于分次Jacobson根和分次素根的关系，同时，给出了分次情形的Cutting down定理和Lying over定理．     

A FINITE STRUCTURE THEOREM BETWEEN PRIMITIVE RINGS AND ITS APPLICATION TO GALOIS THEORY

设\[\mathfrak{M} = \sum {F{u_i}} \]是除环F上向量空间，P是F的一个子除环且在F中是Galois，即存 在F的一个自同构群G使\[I(G) = P\].记Ф是F的中心，\[{G_0}\]是属于G的内自同构群， \[{G_0}\]的元素记为\[{I_r},r \in F\]；，记\[{E^'} = \sum\limits_{{I_{{r_j}}} \in {G_0}} {{\Phi _{{r_j}}}} \]是G的代数，\[P' = {C_F}({E^'})\]是\[{E^'}\]在F中的中心化子.记\[\mathfrak{U}(F,\mathfrak{M})\]是\[\mathfrak{M}\]的F-线性变换完全环，\[{T_v}(F,\mathfrak{M})\]是\[\mathfrak{U}(F,\mathfrak{M})\]中所有秩小于\[\mathcal{X}{_v}\]的元素集合，那末我们有如下主要结果： (1)\[{[F:P']_L} = n\]有限当且仅当\[{T_v}(P',\mathfrak{M}) = \sum\limits_{j = 1}^n \oplus {r_{jL}}{T_v}(F,\mathfrak{M})\]，其中\[{r_j} \in {E^'}\]，\[{r_{jL}}\]表示元素\[{r_j}\]的标量左乘. （2）\[{[P':P]_L} = t\]有限当且仅当凡\[{T_v}(P,\mathfrak{M}) = \sum\limits_{j = 1}^t \oplus {S_j}{T_v}(P',\mathfrak{M})\],其中\[{S_j}\]表示\[\mathfrak{M}\]的F-半线变换自同构，它的伴随同构\[{\psi _j} \in G\]. ⑶如有某个序数v使\[{T_v}(P,\mathfrak{M})\]，\[{T_v}(P',\mathfrak{M})\]及\[{T_v}(F,\mathfrak{M})\]满足⑴及(2)中的关系 式，那末对任何\[{T_\mu }(P,\mathfrak{M})\]，\[{T_\mu }(P',\mathfrak{M})\]及\[{T_\mu }(F,\mathfrak{M})\]皆满足(1)及(2)中的关系式.特别 对\[\mathfrak{U}(P,\mathfrak{M})\]，\[\mathfrak{U}(P',\mathfrak{M})\]及\[\mathfrak{U}(F,\mathfrak{M})\]也是如此. ⑷如果\[{[F:P]_L}\]有限，那末必有\[{C_p}({C_F}(E')) = E'\]，\[{[F:P']_L} = \dim E'\]，\[{[P':P']_L} = [G/{G_0}]\]，其中dim E'表示E'在\[\Phi \]上的维数，\[[G/{G_0}]\]表示\[{G_0}\]在G中的指数,特别\[G\]是 Galois 群，则 \[{C_F}(P') = {C_F}(P) = E'\]. (5)若\[{\tilde G}\]是F的另一自同构群且\[I(G) = I(\tilde G)\]，那末必有\[[G/{G_0}] = [\tilde G/{{\tilde G}_0}]\]， \[\dim {\kern 1pt} {\kern 1pt} E' = \dim {\kern 1pt} {\kern 1pt} \tilde E'\]. 其中\[{\kern 1pt} \tilde E'\]表示\[{\tilde G}\]的代数. 如果P取为F的中心时，于是从上述结果(1)就得出熟知的定理：\[[F:\Phi ]\]是有限的当 且仅当\[\mathfrak{U}(\Phi ,\mathfrak{M}) = \mathfrak{U}(F,\mathfrak{M}){ \otimes _\Phi }{F_L}\]. 另方面，运用我们上述的结果，可导出除环F的有限Galois理论.     

GRADED RINGS OF LOCAL FINITE REPRESENTATION TYPE

Abstract

Graded Artin algebras whose category of graded modules is locally of finite representation type are introduced. The representation theory of such algebras is studied. In the hereditary case and in the stably equivalent to hereditary case, such algebras are classified.     

与线性变换的完全环同构的环理论(Ⅰ)

<正> 熟知地,满足极小条件的单纯环只与一个有限维向量空间的线性变换的完全环同构.并且此向量空间如取为左向量空间的话,那末R的任一极小右理想均可取为此左向量空间.在没有有限条件情况下,Jacobsoo用本原环来取代这种单纯环.接着Wolfson研     

与线性变换的完全环同构的环理论(Ⅲ)

<正> 本文继上文[1,2]的理论,对线性变换完全环的结构作进一步研究.在§1中我们讨论一般无限矩阵的几何意义.在§2中我们用有限维向量空间的线性变换完全环来构作无限维向量空间的线性变换完全环.我们的思想方法是:设是向量空间,     

与线性变换的完全环同构的环理论(Ⅳ)

<正> 基座概念对本原环的结构研究起着十分重要作用.为了对本原环的结构作进一步研究,我们引进俨基座概念.通常基座概念就是我们特殊情形的o-基座概念.利用ν-基座概念,我们建立了ν-结构定理。通常本原环结构定理(见[2]p.75)是ν-结构定理的一种特殊情况. 为了引进ν-基座,我们改变一下本原环的基座定义,使它具有能表达ν-基座的一般形式的特点且能建立所要求的ν-结构定理.为此我们来提一下§2中所获得的结果:     

与线性变换的完全环同构的环理论(Ⅱ)

<正> 为了进一步对本原环结构的研究,本文引进规范环的概念,我们说环R是规范的,若R是一个线性变换完全环并且及的基座对于任一对应基{E_i}皆有=∑RE_i=∑E_iR.容易知道,满足单侧理想极小条件的单纯环必是规范的.     

ASYMPTOTIC PROPERTIES OF ELEMENTS IN GROUP RINGS OF FINITE GROUPS

A relation between algebraic degrees of eigenvalues of an element in the group ring Z G of a finite group G and their multiplicities is obtained. This result can be regarded as a theorem on asymptotic behavior of parameters in group rings of finite groups which is a parallel result of several well known theorems concerning the relationships of parameters in finite groups.     

LOCALLY STABILIZED FINITE ELEMENT METHOD FOR STOKES PROBLEM WITH NONLINEAR SLIP BOUNDARY CONDITIONS

Based on the low-order conforming finite element subspace （Vh, Mh） such as the P1-P0 triangle element or the Q1-P0 quadrilateral element, the locally stabilized finite element method for the Stokes problem with nonlinear slip boundary conditions is investigated in this paper. For this class of nonlinear slip boundary conditions including the subdifferential property, the weak variational formulation associated with the Stokes problem is an variational inequality. Since （Vh, Mh） does not satisfy the discrete inf-sup conditions, a macroelement condition is introduced for constructing the locally stabilized formulation such that the stability of （Vh, Mh） is established. Under these conditions, we obtain the H1 and L2 error estimates for the numerical solutions.     

有限級有序加羣及有序環的表現

在另一文中,我们讨论了由全体2维實向量所成的有序环,在该文最後並说當维数n>2时(n为有限)也可类似地作初步讨论.为了显示这种向量环的用途,我们考虑用向量环来表现一般有序环的问题.在本文中我们证明:任一“n级的”(见以下定义)有序环都能与一个由若干n维實向量所组成的有序环同构.(主要在於证出关於n级有序加羣的类似结果.)我们希望有较好的结果,即:任一n级有序环都能与由全体n维實向量所成的一个有序环的一个子环同构,但未能证明或否定.     

抛物型方程非齐次边值问题的推广型LOD有限差分及有限元格式

1引言本文考虑区域Ω=[0,1]~d(d=2,3)上的非齐次抛物型方程第一边值问题(?)-C_1△u C_2u=f(x,t),x∈Ω,t∈(0,T],(1．1) u(x,0)=u_0(x),x∈Ω,(1．2) u(x,t)=(?)(x,t),x∈(?)Ω,t∈(0,T],(1．3)其中C_1,C_2为常数且C_1＞0,C_2≥0．对于以上问题,可以使用有限差分方法及有限元方法进行离散,并采用交替方向方法求解．交替方向方法能够将高维问题转化为一系列的一维问题进行计算,具有计算量少,计算稳定且易于并行实现等优点,在大规模科学计算中起着非常重要的作用,一直是计算数