关于斜McCoy环

For a ring endomorphism α,we introduce α-skew McCoy rings which are generalizations of α-rigid rings and McCoy rings,and investigate their properties.We show that if α t = I R for some positive integer t and R is an α-skew McCoy ring,then the skew polynomial ring R[x;α] is α-skew McCoy.We also prove that if α（1） = 1 and R is α-rigid,then R[x;α]/ x 2 is αˉ-skew McCoy.     

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.     

Linearly McCoy Rings and Their Generalizations

A ring R is called linearly McCoy if whenever linear polynomials f（x）, g（x） e R[x]／{0） satisfy f（x）g（x） ： O, then there exist nonzero elements r, s ∈ R such that f（x）r ： sg（x） =0. For a ring endomorphism α, we introduced the notion of α-skew linearly McCoy rings by considering the polynomials in the skew polynomial ring R[x; α] in place of the ring R[x]. A number of properties of this generalization are established and extension properties of α-skew linearly McCoy rings are given.     

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.     

On Skew Triangular Matrix Rings

Let α be a nonzero endomorphism of a ring R, n be a positive integer and T_n(R, α) be the skew triangular matrix ring. We show that some properties related to nilpotent elements of R are inherited by T_n(R, α). Meanwhile, we determine the strongly prime radical, generalized prime radical and Behrens radical of the ring R[x; α]/(x~n), where R[x; α] is the skew polynomial ring.     

拟-Armendariz模

For a right R-module N, we introduce the quasi-Armendariz modules which are a common generalization of the Armendariz modules and the quasi-Armendariz rings, and investigate their properties. Moreover, we prove that NR is quasi-Armendariz if and only if Mm(N)Mm(R) is quasi-Armendariz if and only if Tm(N)Tm(R) is quasi-Armendariz, where Mm(N) and Tm(N) denote the m×m full matrix and the m×m upper triangular matrix over N, respectively. NR is quasi-Armendariz if and only if N[x]R[x] is quasi-Armendariz. It is shown that every quasi-Baer module is quasi-Armendariz module.     

Generalized PP and Zip Subrings of Matrix Rings

Let R be an abelian ring. We consider a special subring An, relative to α2,…, αn∈ REnd（R）, of the matrix ring Mn（R） over a ring R. It is shown that the ring An is a generalized right PP-ring （right zip ring） if and only if the ring R is a generalized right PP-ring （right zip ring）. Our results yield more examples of generalized right PP-rings and right ziu rings.     

Principal quasi-Baerness of formal power series rings

Let R be a ring. We consider left （or right） principal quasi-Baerness of the left skew formal power series ring R[[x;α]] over R where a is a ring automorphism of R. We give a necessary and sufficient condition under which the ring R[[x; α]] is left （or right） principally quasi-Baer. As an application we show that R[[x]] is left principally quasi-Baer if and only if R is left principally quasi- Baer and the left annihilator of the left ideal generated by any countable family of idempotents in R is generated by an idempotent.     

矩阵环中的一类极大的广义的Armendariz子环

An associative ring with identity R is called Armendariz if, whenever （∑^m i=0^aix^i）（∑^n j=0^bjx^j）=0 in R[x],aibj=0 for all i and j. An associative ring with identity is called reduced if it has no non-zero nilpotent elements. In this paper, we define a general reduced ring （with or without identity） and a general Armendariz ring （with or without identity）, and identify a class of maximal general Armendariz subrings of matrix rings over general reduced rings.     

A Note on Pseudo Ideals of Fields

Definition A subring A of a ring R is called a pseudo ideal, if for any r∈R and α∈A, r~2α∈A and αr~2∈A. The subring (R)~2={Σ±r_1~2…r_i~2|r_i∈R, t∈N} is called the square closed ring of R. Lemma The square closed ring of a division ring D is the intersection of all nontrivial pseudo ideals of D. Theorem Ⅰ If a simple ring R with a unit has a nontrivial pseudo ideal A containing an invertible element, R has characteristic p=2. In particular, the necessary condition for a division ring containing a nontrivial pseudo ideal is     

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.     

$2$-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.     

On Centralizer Subalgebras of Group Algebras

Let $G$ be a finite group, $H ≤ G$ and $R$ be a commutative ring with an identity $1_R$. Let $C_{RG}(H) = \{ α ∈ RG|αh= hα$ for all $h ∈ H \}$, which is called the centralizer subalgebra of $H$ in $RG$. Obviously, if $H = G$ then $C_{RG}(H)$ is just the central subalgebra $Z(RG)$ of $RG$. In this note, we show that the set of all $H$-conjugacy class sums of $G$ forms an $R$-basis of $C_{RG}(H)$. Furthermore, let $N$ be a normal subgroup of $G$ and $γ$ the natural epimorphism from $G$ to $\overline{G}= G/N$. Then $γ$ induces an epimorphism from $RG$ to $R\overline{G}$, also denoted by $γ$. We also show that if $R$ is a field of characteristic zero, then $γ$ induces an epimorphism from $C_{RG}(H)$ to $C_{R\overline{G}}(\overline{H})$, that is, $γ(C_{RG}(H)) = C_{R\overline{G}}(\overline{H})$.     

矩阵环的半交换子环

称环R是半交换的,如果对任意a∈R,rR(a)是R的理想．若n≥2,则任意具有单位元的环R上的n阶上三角矩阵环不是半交换环．我们证明了reduced环上的上三角矩阵环的一类特殊子环是半交换环．     

${\mbox{\boldmath $R$}}^N$上奇异非线性多调和方程的正整体解

本文研究形如△((△nu)(p-1) )=f(|x|,u,|(?)u|)u-β,x∈RN的奇异非线性多调和方程在RN上的正整体解,此处P>1,β≥0是常数,n是自然数,f:R × R ×R →R 是一个连续函数, ξδ*:=sign(ξ)·|ξ|δ,,ξ∈R,δ>0,给出了该类方程具有无穷多个其渐进阶刚好为|x|2n的正整体解的充分条件与必要条件.这些结论可以推广到更一般的方程.     

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．     

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）.