“关于除环的伪理想”一文的更正

对于一般的有限域F,设|F|=P~m,p为素数,m∈N,如所周知,F是其素子域Z_p的单代数扩张:F=Z_p(u),F有无真n-伪理想,决定于代数元u的质式P(x)∈Z_p〔x〕的结构。我们知道,P(x)|x~(p~m)-1,而x~(p~m)-1=(x-1)(x~(p~m-2)+x~(p~m-3)+…+x+1),故P(x)|x-1,或P(x)|x~(p~m-2)+x~(p~m-3)+…x+1。对于前一种情形,P(x)=x-1,u=1,F=Z_p,已由定理6所讨论。对于后一种情形,只知道P(x)是x~(p~m-2)+x~(p~m-3)+ …+x+1的因子,直接由P、m和n给出F无真n-伪理想的充要条件是不可能的,它需要具体地知道质式P(x)的结构才能作出判断。但是我们有     

可除半环的伪理想与同余

引入可除半环的伪理想,它与同余互相唯一决定,在可除半环中利用理想作商是可行的,并证明了同态基本定理.     

关于正则环的理想

本文把对于正则环的可比较性推广到对于正则环上理想的可比较性,得到这 种可比较性的若干刻画.进而我们证明,对稳定秩1的理想, Roth问题有肯定解答.     

关于Γ-除环

文中首先给出了定理２．２,由它可以行到文献「４」和「５」中关于Γ－域和除Γ－环的一些结果,其次,我们还指出了一个Γ－环是Γ－除环的几个条件。     

关于循环环的理想、素理想和极大理想

研究了循环环R=的理想、素理想和极大理想的个数和结构,得到了如下结论:1)理想:(1)若|R|=∞,则R共有无穷多个理想:;(2)若|R|=n,设n的正因数个数为T(n),则R共有T(n)个理想:.2)素理想:(1)若|R|=∞,设a^2=ka(k≥0),①当k=0时,R的素理想只有R;②当k>0时,R的素理想共有无穷多个,它们是:{0}、R及;(2)若|R|=n>1,设a^2=ka,0≤k.3)极大理想:(1)若|R|=∞,则R有无限多个极大理想,它们是;(2)若|R|=n>1,设n的互不相同的素因数个数为ψ(n),则R共有ψ(n)个极大理想:(pa|p是n的素因数).     

除环上的全阵环的极小右理想与半素F-环

说环R是F-环,如R含一有限非零元集X,使对任意α∈R,若αR≠0,则αR∩X≠φ(傅昶林)。半素F-环可表为有限个除环上的全阵环的直和(周毅强)。有人指出,这个命题的逆命题是不对的,今给出环为半素F-环的充要条件,先看除环上的全阵环。 设D为一除环,n>1为一自然数,R为D上n阶全阵环。极小右理想均为主右理想、取α=(α_(ij))≠0∈R,设其中某α_(ij)≠0∈D,则     

关于环的极大本质右理想

设Ｒ为环，我们考虑下面两个条件。（＊）Ｒ的每个极大本质右理想是ＧＰ－内射右Ｒ－模或右零化子．（＊）Ｒ的每个极大本质右理想是ＹＪ－内射右Ｒ－模．本文旨在研究满足条件（＊）或（＊）的环，同时我们还给出了强正则环和除环的一些新刻画．     

关于除环上分块矩阵秩的等式

本文使用双矩阵分解方法研究除环上分块矩阵秩的等式,给出了Marsaglia-Styan公式一个新的证明并获得了一些新的秩等式的刻划.     

关于p除环结构的几个定理

本文讨论了强ｐ除环的结构与赋值，并且给出了ｐ除环是广义四元数体的一个子体的充要条件。     

On ideals of triangular matrix rings

We provide a formula for the number of ideals of complete block-triangular matrix rings over any ring R such that the lattice of ideals of R is isomorphic to a finite product of finite chains, as well as for the number of ideals of (not necessarily complete) block-triangular matrix rings over any such ring R with three blocks on the diagonal.     

On Jordan ideals and matrix rings

For A, a commutative ring, and results by Costa and Keller characterize certain -normalized subgroups of the symplectic group, via structures utilizing Jordan ideals and the notion of radices. The following work creates a Jordan ideal structure theorem for -graded rings, A₀A₁, and a -graded matrix algebra. The major theorem is a generalization of Costa and Keller’s previous work on matrix algebras over commutative rings.     

On strongly prime rings and ideals

Strongly prime rings may be defined as prime rings with simple central closure. This paper is concerned with further investigation of such rings. Various characterizations, particularly in terms of symmetric zero divisors, are given. We prove that the central closure of a strongly (semi-)prime ring may be obtained by a certain symmetric perfect one sided localization. Complements of strongly prime ideals are described in terms of strongly multiplicative sets of rings. Moreover, some relations between a ring and its multiplication ring are examined.     

On the normal ideals of exchange rings

An ideal I of a ring R is called normal if all idempotent elements in I lie in the center of R. We prove that if I is a normal ideal of an exchange ring R then: (1) R and R/I have the same stable range; (2) V(I) is an order-ideal of the monoid C(Specc(R), N), where Specc(R) consists of all prime ideals P such that R/P is local.