具有对称自同态的环及其扩张

Let R be a ring with an endomorphism α and an α-derivation δ. We introduce the notions of symmetric α-rings and weak symmetric α-rings which are generalizations of symmetric rings and weak symmetric rings, respectively, discuss the relations between symmetricα-rings and related rings and investigate their extensions. We prove that if R is a reduced ring and α(1) = 1, then R is a symmetric α-ring if and only if R[x]/(x n) is a symmetric ˉα-ring for any positive integer n. Moreover, it is proven that if R is a right Ore ring, α an automorphism of R and Q(R) the classical right quotient ring of R, then R is a symmetric α-ring if and only if Q(R) is a symmetric ˉα-ring. Among others we also show that if a ring R is weakly 2-primal and(α, δ)-compatible, then R is a weak symmetric α-ring if and only if the Ore extension R[x; α, δ] of R is a weak symmetric ˉα-ring.     

拟-中心半交换环

环$R$称为拟-中心半交换的(简称QCS环)如果对$a,b\in R$, $ab=0$蕴含$aRb\subseteq Q(R)$, 其中$Q(R)$为$R$的拟中心.证明了如果$R$ 为QCS环, 那么$R$的幂零元集恰好是它的Wedderburn根, 且对$n\geq 2$, 上三角矩阵环$R=T_n(S)$ 是QCS 环当且仅当$n=2$ 且$S$ 是duo 环, 而$T_{2k+2}^k$是QCS环如果$R$是约化的duo环.     

关于w-linked扩环

Let R ■ T be an extension of commutative rings.T is called w-linked over R if T as an R-module is a w-module.In the case of R ■ T ■ Q 0 （R）,T is called a w-linked overring of R.As a generalization of Wang-McCsland-Park-Chang Theorem,we show that if R is a reduced ring,then R is a w-Noetherian ring with w-dim（R） 1 if and only if each w-linked overring T of R is a w-Noetherian ring with w-dim（T ） 1.In particular,R is a w-Noetherian ring with w-dim（R） = 0 if and only if R is an Artinian ring.     

斜Armendariz矩阵环

Let R be a ring.We show in the paper that the subring Un(R) of the upper triangular matrix ring Tn(R) is α-skew Armendariz if and only if R is α-rigid,also it is maximal in some non α-skew Armendariz rings,where α is a ring endomorphism of R with α(1) = 1.     

半交换环的一个推广研究

In this paper, a generalization of the class of semicommutative rings is investigated.A ring R is called left GWZI if for any a ∈ R, l(a) is a GW-ideal of R. We prove that a ring R is left GWZI if and only if S3(R) is left GWZI if and only if Vn(R) is left GWZI for any n ≥ 2.     

交换环上矩阵代数的局部化导子

Let R be a commutative ring with identity, Nn（R） the matrix algebra consisting of all n × n strictly upper triangular matrices over R. Several types of proper local derivations of Nn（R） （n ≤ 4） are constructed, based on which all local derivations of Nn（R） （n ≤ 4） are characterized when R is a domain.     

广义IP-内射环

对环R,令ip(R_R)={a∈R：任意一个从R的右理想到R且象为aR的模同态能开拓到R}。众所周知,R为右IP-内射环当且仅当R=ip(R_R),R为右单-内射环当且仅当{a∈R：aR is simple)(?)ip(R_R)。对环R的一个子集S,我们引进了S-IP-内射环的概念,即满足S(?)ip(R_R)的环。并得到了这种环的一些性质。     

非交换环的拟零因子图

In this paper, a new class of rings, called FIC rings, is introduced for studying quasi-zero-divisor graphs of rings. Let R be a ring. The quasi-zero-divisor graph of R, denoted by Γ_*(R), is a directed graph defined on its nonzero quasi-zero-divisors, where there is an arc from a vertex x to another vertex y if and only if x Ry = 0. We show that the following three conditions on an FIC ring R are equivalent:(1) χ(R) is finite;(2) ω(R) is finite;(3)Nil_*R is finite where Nil_*R equals the finite intersection of prime ideals. Furthermore, we also completely determine the connectedness, the diameter and the girth of Γ_*(R).     

导子交换子幂值的零化子条件

Let R be a prime ring with center Z(R), I a nonzero ideal of R, d a nonzero derivation of R and 0≠a∈ R. In the present paper, our object is to study the situation a[d(xk), xk]n∈ Z(R) for all x ∈ I under certain conditions, where n(≥ 1), k(≥ 1) are fixed integers.     

$2$-Primal环的Ore扩张中的极小素理想和单位元

Let R be an(α,δ)-compatible ring.It is proved that R is a 2-primal ring if and only if for every minimal prime ideal P in R[x;α,δ] there exists a minimal prime ideal P in R such that P = P [x;α,δ],and that f(x) ∈ R[x;α,δ] is a unit if and only if its constant term is a unit and other coefficients are nilpotent.     

关于 Radicaltotal 环上的投射模

文中给出了Radicaltotal环上投射模的分解定理. 对一个 ~uniform 维数有限的 Totalfree 环 R, 该文证明 R 是一个总体维数≤ 1 的诺特环, 且 R上的任何投射模必同构于$ \bigoplus\limits_{i\in I}Re_{i}$, 其中每个 $e_{i}$ 均为 $R$ 的非零幂等元. 此外, 文中还给出了一些相关的例子.     

关于素环平方封闭的李理想的两个结果

Let R be a 2-torsion free prime ring, d1 a nonzero derivation, -γ a generalized derivation associated with a nonzero derivation d2, U a square closed Lie ideal of R. In the present paper,we prove that if [di^2（u）, u] ∈ Z（R） or γ acts as a homomorphism （or an antihomomorphism） on U, then U Z（R）.     

关于剩余类环的扩展的研究

作者对非结合环给出扩展的概念,即给定2个非结合环A和B,对任一非结合环R,称R是A被B的扩展,当且仅当A是R的理想且R/A≌B.对非结合环的扩展,文中证明了一个类似于Schreier群扩张定理的结果.作为应用,对给定的自然数m≥2,n≥2,文章刻画了模n的剩余类环Z_n被模m的剩余类环Z_m扩展所得到的有限环R的构造,证明了R可以用满足一定条件的自然数对(u,r)来描述,同时写出了R的理想和单侧理想的具体形状.作者还进一步证明,R是结合的当且仅当R=Z_nZ_m,且当R=Z_nZ_m时,R的每个理想都是Z_n的一个理想与Z_m的一个理想的直和,即此时R的理想是相对平凡的.     

THE STRUCTURE OF ORTHOGONAL GROUPS OVER ARBITRARY COMMUTATIVE RINGS

Let R be an arbitrary commutative ring, and n an integer≥3. It is proved for any ideal J of R thatEO_(2n)(R, J)=[EO_(2n)(R), EO_(2n)(J)]=[EO_(2n)(R), EO_(2n)(R, J)]=[EO_(2n)(R), O_(2n)(R,J)]=[O_(2n)(R), EO_(2n)(R,J)].In particular, EO_(2n)(R, J) is a normal subgroupof O_(2n)(R). Furthermore, the problem of normal subgroups of O_(2n)(R) has an affirmative solution if and only if aR∩ Ann(2)=α~2 Ann(2) for each a in R. In particular, if 2 is not a zero divisor in R, then the problem of normal subgroups of O_(2n)(R) has an affirmative solution     

可换环上矩阵代数的SZ-导子, PZ-导子 和 S-导子

Let R be a commutative ring with identity, Nn（R） the matrix algebra consisting of all n × n strictly upper triangular matrices over R with the usual product operation. An R-linear map φ ： Nn（R） → Nn（R） is said to be an SZ-derivation of Nn（R） if x2 = 0 implies that φ（x）x＋xφ（x） = 0. It is said to be an S-derivation of Nn（R） if φ（x2） = φ（x）x＋xφ（x） for any x ∈ Nn（R）. It is said to be a PZ-derivation of Nn（R） if xy = 0 implies that φ（x）y＋xφ（y） = 0. In this paper, by constructing several types of standard SZ-derivations of Nn（R）, we first characterize all SZ-derivations of Nn（R）. Then, as its application, we determine all S-derivations and PZ- derivations of Nn（R）, respectively.     

形式三角矩阵环的PS性质和CESS性质

设$R$是环. 称右$R$-模$M$是PS-模,如果$M$具有投射的socle. 称$R$是PS-环,如果$R_R$是PS-模. 称$M$是CESS-模,如果$M$的任意具有基本socle的子模是$M$的某个直和因子的基本子模.本文给出了形式三角矩阵环 $T=\left( \begin{array}{cc} A & 0 \\     

关于Baer PP和PS环的一些性质(英)

在此文中,我们对Strong-Armendariz环和Baer PP及PS环Ore-扩张R[x,x~(-1)；α]的一些性质进行了讨论研究,并得到了一些结果．主要证明了R是Baer(PP)环当且仅当R[[x]]是Baer(PP)环及R是α-rigid环时,R是Baer(PP,PS)环当且仅当R[[x]]是Baer(PP,PS)环．     

一类上三角矩阵环$W_{n}(p, q)$的斜Armendariz性质

设α是环R的一个自同态,称环R是α-斜Armendariz环,如果在R[x;α]中,(∑_(i=0)~ma_ix~i)(∑_(j=0)~nb_jx~j)=0,那么a_ia~i(b_j)=0,其中0≤i≤m,0≤j≤n.设R是α-rigid环,则R上的上三角矩阵环的子环W_n(p,q)是α~—-斜Armendariz环.     

On Jordan Biderivations of Triangular Matrix Rings

Let $R$ and $S$ be rings with identity, $M$ be a unitary $(R,S)$-bimodule and $T=\left(\begin{array}{cc}R & M \\ 0 & S\end{array}\right) $ be the upper triangular matrix ring determined by $R$, $S$ and $M$. In this paper we prove that under certain conditions a Jordan biderivation of an upper triangular matrix ring $T$ is a biderivation of $T$.