唯一分解整环上多项式不可约的一个判别法

给出了唯一分解整环上多项式不可约的一个判别法.     

关于不同因子分解的数目

设f(n)表示分解自然数n(>1)为大于1的整数因子乘积的所有方式的数目(不计因子的顺序),并设0<β<1,N(x,β)=Card{n≤x,f(n)≥n~β}.本文分别估计了N(x,β)和f(n))的值.     

Z[√-m]为唯一因子分解整环的刻画

利用一类整环Z[√-m]={x＋y√-m｜x,y∈z,m∈N}中不可约元的几个性质,给出了这类整环成为唯一因子分解整环的一个充分必要条件．     

用矩阵的初等行变换求欧氏环中元素的最大公因子

本文给出了欧氏环中多个元素的最大公因子的矩阵求法,解决了求欧氏环上的n元一次不定方程的所有解问题。设R为欧氏环,a_i∈R(i=1,2,…,),则a_1,a_2,…,a_n的最大公因子d=(a_1,a_2,…,a_n)—     

完全图的素圈因子分解

图的因子理论是图论研究中最活跃的课题之一.其中心问题是把一个图分解成具有给定性质的因子.     

有单位元交换环上二阶矩阵的因子分解

对有单位元交换环上矩阵分解问题进行了讨论,给出了有单位元交换环上二阶矩阵可以因式分解的充分必要条件,即单位元交换环上二阶矩阵可以因式分解当且仅当这个矩阵的行列式可以因子分解.     

欧氏环上矩阵的标准形

将线性变换放至环上线性模中来研究 ,并给出了欧氏环上矩阵的标准形 .     

一个特征标环素谱的连通性

本文确定了一个有限群特征标环通过代数整数环扩张后素谱的结构，在此基础之上，利用与[1]中类似的方法证明了这个扩张后的特征标环素谱的连通性。同时还计算了有限群的复类函数空间的幂等元．     

PRIME AND MAXIMAL IDEALS IN CONSTRUCTIVE RING THEORY

The basic notions of ideal theory are examined constructively in the context of a commutative ring with an identity and an inequality relation. Constructive analogues of classical theorems relating maximality and primeness are proved, and it is shown that the results are the best possible in a constructive framework.     

ON PRIME IDEALS OF THE RING OF DIFFERENTIAL OPERATORS

For given linear differential operators \[{P_1}(\frac{\partial }{{\partial {x_1}}},...,\frac{\partial }{{\partial {x_n}}}),...,{P_n}(\frac{\partial }{{\partial {x_1}}},...,\frac{\partial }{{\partial {x_n}}})\], let \[({P_1},...,{P_m})\] be the left ideal generated by \[{P_1},...,{P_m}\], and F be the space of the common solutions of the operators \[{P_i}(i = 1,...,m),i,e.{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} F = \{ f|{P_i}f = 0,i = 1,...,m\} \].Inspired by the Hilbert's Nullstailerisatz, we introduce another ideal \[H({P_1},...,{P_m})\] related to \[{P_1},...,{P_m}\], \[H({P_1},...,{P_m}) = \{ P|Pf = 0,\forall f \in F\} \], Obviously, \[({P_1},...,{P_m}) \subseteq H({P_1},...,{P_m})\]. Definition The ideal \[({P_1},...,{P_m})\] is prime if \[H({P_1},...,{P_m}) = ({P_1},...,{P_m})\]. The aim of this paper is to stndy under what conditions the ideal \[({P_1},...,{P_m})\] is prime. We have the following results： Theorem 1 \[(\Delta )\]is prime, where \[\Delta \] is the Laplace operator of \[{{\cal R}^n},i,e\], \[\Delta = \sum\limits_{i = 1}^n {\frac{{{\partial ^2}}}{{\partial x_i^2}}} \] Theorem 2 \[{\Delta _S}\] is prime, where \[\Delta \] is the Laplace-Beltrami operator of \[B = \{ ({Z_1},...,{Z_n}) \in {{\cal L}^n}|\sum\limits_{i = 1}^n {|{z_i}} {|^2} < 1\} \] it is well hmwn that \[{\Delta _S} = \sum\limits_{i,j = 1}^n {({\delta _{ij}} - {z_i}{{\bar z}_j})\frac{{{\partial ^2}}}{{\partial {z_i}\partial {{\bar z}_j}}}} \] Theorem 3 Suppose \[{T_1}(\frac{\partial }{{\partial {x_1}}},...,\frac{\partial }{{\partial {x_n}}}),...,{T_m}(\frac{\partial }{{\partial {x_1}}},...,\frac{\partial }{{\partial {x_n}}})\] are differential operators with constant coefficients, then \[({T_1},...,{T_m})\] is prime if the corresponding polynomial ideal \[({T_1}({X_1},...,{X_n}),...,{T_m}({X_1},...,{X_n}))\] is prime in the usual sense. For general linear differential operators with variable coeffcients, we have given some sufficient conditions for which \[({P_1},...,{P_m})\] is prime (see Th. 4). The two important ones among these sufficient conditions are that \[{P_1},...,{P_m}\] possess a Poisson kernel and they satisfy group invariance in some sense. Finally, as an example, we study in detail the case of the classical domain \[R(2) = \{ Z = \left( {\begin{array}{*{20}{c}} {{z_1}}&{{z_2}}\{{z_3}}&{{z_4}} \end{array}} \right)|I - Z{Z^'} > 0\} \] We show that \[H({\Delta _{R(2)}}),i,e,({\Delta _{R(2)}})\] is not prime, and we also give the basis of the ideal \[H({\Delta _{R(2)}})\], where \[H({\Delta _{R(2)}})\] is the Laplaoe-Beltrami operator of R(2).     

ZEROS OF CHARACTERS ON PRIME ORDER ELEMENTS

Suppose that G is a finite group, let χ be a faithful irreducible character of degree a power of p and let P be a Sylow p-subgroup of G. If χ(x) ≠ 0 for all elements of G of order p, then P is cyclic or generalized quaternion.

     

具有很多素数阶元的有限群

考察元素的阶如何影响有限群的结构是群论中的一个重要课题．本文研究存在一个正规子群N，N外的元素都是素数阶元的有阶群．主要利用熟知的Thompson的一个定理，获得了这样的有限群．     

本原环的秩等于1的幂等元的分类及其应用

本文提出了含非零基座本原环的全体秩等于１的幂等元的两种分类，据此进一步提示了此类本原环的结构。     

有限群中素数方幂阶子群的个数

本文利用基础代数中有关稳定子、陪集等理论,给出了有限群G中pk阶子群个数的 一个结果.     

关于四维Liouville方程解的注记

本文求得了四维Euclid空间中liouville方程的依赖于两个任意复Fueter解析函数的一般解,并讨论了一般情形.     

複合形在歐氏空間中的實現問题Ⅰ

<正> 在拓撲發展之初很早就知道一個抽象的n維單純複合形(有限或無限)必可在2n+1維歐氏空間及R~(2n+1)中得到實現,它的證明也很簡單(例如見[1]§2或[2]第Ⅲ章§2).從這一定理知道2n+1維的歐氏空間實際上已包括了所有想像得到的n維複合形,可是是否有不能在R~m中實現但能在R~(m+1)中實現的