关于素环导子的一些结论

设R是一个特征不等于2的不可交换的素环,d为R的一个导子,如果[xd,x]xd=0 对所有的x∈R都成立,那么d=0.进一步,如果对所有的x∈R,都有[[xd,x],xd]=0,那么d=0.     

素环上的导子在Lie理想上的协交换子的零化子

Let R be a prime ring of characteristic different from 2,d and g twoderivations of R at least one of which is nonzero,L a non-central Lie ideal of R,anda∈R.We prove that if a(d(u)u-ug(u))=0 for any u∈L,then either a=0,or R is an s_4-ring,d(x)=[p,x],and g(x)=-d(x)for some p in the Martindalequotient ring of R.     

半素环上的微商

本文讨论了微商作用在半素环的某些左理想上的问题.给出了如下结果:设R是带有中心Z（R）的半素环.d和g是R的微商,L为R的非零左理想且r_R（L）=0.假设d（x）x-xg（x）∈Z（R）对任意的x∈L成立.那么d（R）?Z(R）且由d（R）生成的R的理想在R的中心里.     

具有“积分小”系数的中立型方程的振动性

讨论了中立型方程d/(dt)[x(t) - R(t)x(t - r)] + P(t)x(t - τ) - Q(t)x(t - δ) = 0,的振动性,其中P,Q,R∈C（[t₀,∞), R⁺）,r,τ,δ∈（0,∞）,得到若干新结果。     

素环的李结构

设R是一个咎征非2的素环，U是R的一个平方封闭的李理想，d1，d2，d是R的导子，δ是R的广义导子．本文证明了U为中心李理想，如果以下条件之一成立：（1）d（x）od（y）=xoy；（2）d（x）οd（y）＋xοy=0；（3）d1（x）οd2（可）=0；（4）δ（[x，y]）=0；（5）δ（xοy）=0对所有的x，y∈U．     

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.     

Generalized Derivations with Nilpotent Values on Semiprime Rings

Let R be a semiprime ring, RF be its left Martindale quotient ring and I be an essential ideal of R. Then every generalized derivation μ defined on I can be uniquely extended to a generalized derivation of RE. Furthermore, if there exists a fixed positive integer n such that μ(x)^n = 0 for all x∈I, then μ=0.     

环的广义斜导子

设R是一个半素环, RF(resp．Q)是它的左Martindale商环(对称Martindale 商环),K是R的一个本质理想,则K上的每一个广义斜导子μ能被唯一地扩展到RF和Q 上．设R是一个素环,K是R的一个本质理想,μ是K上的一个广义斜导子且α为其伴随自同构,d为其伴随斜导子,如果存在n≥0,使得对任意的x∈K都有μ(x)n=0,那么μ=0．     

关于周期环与Jacobson环的几个定理

A ring R is called a periodic ring, if for every x∈R there exist two distinct positive integers m(x) and n(x) such that x^m(x)=x^n(x)(cf. [1]), in particularly, if m(x) = 1 for any x∈R. then this periodic ring is called a Jacobson ring (cf. [2]).In this paper, a necessary and sufficient condition for a ring to be a Jacobson ring is given and some necessary and sufficient conditions for a periodic ring or a Jacobson ring to be a field are also given.     

完全不确定Hamburger矩阵矩量问题的有限阶解

该文描述带有矩量序列{v_m}_0~∞■C~(q×q)的完全不确定Hamburger矩阵矩量问题:v_m=integral from n=-∞to∞x~m dρ(x),m=0,1,…的有限阶解,即该问题的那些解ρ,使得C~(q×q)-值多项式的线性空间P在对应的空间L~2(R,dρ/E(x))内稠密,这里E(x)为在实轴R上取正值的某个数值多项式.作为预备知识,作者考虑所谓广义Akhiezer插值的矩阵变种与它的相关矩阵矩量问题之间的一种关系.     

右对称环

本文在左对称环的基础上提出了右对称环的概念,分别给出了是右对称环但不是左对称环和是左对称环但不是右对称环的例子．证明了（1）如果R是Armendariz环,则R是右对称环的充要条件R[x]是右对称环;（2）如果R是约化环,则R[x]/（x^n）是右对称环,其中（xn）是由xn生成的理想．     

On CS Rings and QF Rings

In this paper, we shall discuss the conditions for a right SC right CS ring to be a QF ring. In particular, we prove that if R is a right SI right CS ring satisfying the reflexive orthogonal condition (*) and if every CS right R-module is -CS, then R is a QF ring.AMS Subject Classification (1991): 16L30 16L60     

环的几种内射性的关系

我们研究了关于广义自内射环(P-内射环,GP-内射环,AP-内射环,单内射环,n-内射环)的一些关系.     

Strong Cleanness of Matrix Rings Over Commutative Rings

Let R be a commutative local ring. It is proved that R is Henselian if and only if each R-algebra which is a direct limit of module finite R-algebras is strongly clean. So, the matrix ring 𝕄 _n (R) is strongly clean for each integer n > 0 if R is Henselian and we show that the converse holds if either the residue class field of R is algebraically closed or R is an integrally closed domain or R is a valuation ring. It is also shown that each R-algebra which is locally a direct limit of module-finite algebras, is strongly clean if R is a π-regular commutative ring.     

Ideal-Symmetric and Semiprime Rings

Lambek extended the usual commutative ideal theory to ideals in noncommutative rings, calling an ideal A of a ring R symmetric if rst ∈ A implies rts ∈ A for r, s, t ∈ R. R is usually called symmetric if 0 is a symmetric ideal. This naturally gives rise to extending the study of symmetric ring property to the lattice of ideals. In the process, we introduce the concept of an ideal-symmetric ring. We first characterize the class of ideal-symmetric rings and show that this ideal-symmetric property is Morita invariant. We provide a method of constructing an ideal-symmetric ring (but not semiprime) from any given semiprime ring, noting that semiprime rings are ideal-symmetric. We investigate the structure of minimal ideal-symmetric rings completely, finding two kinds of basic forms of finite ideal-symmetric rings. It is also shown that the ideal-symmetric property can go up to right quotient rings in relation with regular elements. The polynomial ring R[x] over an ideal-symmetric ring R need not be ideal-symmetric, but it is shown that the factor ring R[x]/xⁿR[x] is ideal-symmetric over a semiprime ring R.     

The McCoy Condition on Ore Extensions

Nielsen [29 Nielsen , P. P. ( 2006 ). Semi-commutativity and the McCoy condition . J. Algebra 298 : 134 – 141 .[Crossref], [Web of Science ®] , [Google Scholar]] proved that all reversible rings are McCoy and gave an example of a semicommutative ring that is not right McCoy. When R is a reversible ring with an (α, δ)-condition, namely (α, δ)-compatibility, we observe that R satisfies a McCoy-type property, in the context of Ore extension R[x; α, δ], and provide rich classes of reversible (semicommutative) (α, δ)-compatible rings. It is also shown that semicommutative α-compatible rings are linearly α-skew McCoy and that linearly α-skew McCoy rings are Dedekind finite. Moreover, several extensions of skew McCoy rings and the zip property of these rings are studied.     

关于C-环

主要讨论了 C-环的结构以及它与其它环类之间的重要关系 ,从而较深刻地刻划这种环     

J-clean环的推广

如果R中每个元素(对应地,可逆元)均可表示为一个幂等元与环R的Jacobson根中一个元素之和,则称环R是J-clean环(对应地,UJ环).所有的J-clean环都是UJ环.作为UJ环的真推广,本文引入GUJ环的概念,研究GUJ环的基本性质和应用.进一步地,研究每个元素均可表示为一个幂等元与一个方幂属于环的Jacobson根的元素之和的环.     

Trivial Ring Extensions Defined by Arithmetical-Like Properties

In this article we investigate the transfer of the notions of elementary divisor ring, Hermite ring, Bezout ring, and arithmetical ring to trivial ring extensions of commutative rings by modules. Namely, we prove that the trivial ring extension R: = A ? B defined by extension of integral domains is an elementary divisor ring if and only if A is an elementary divisor ring and B = qf(A); and R is an Hermite ring if and only if R is a Bezout ring if and only if A is a Bezout domain and qf(A) = B. We provide necessary and sufficient conditions for R = A ? E to be an arithmetical ring when E is a nontorsion or a finitely generated A ? module. As an immediate consequences, we show that A ? A is an arithmetical ring if and only if A is a von Neumann regular ring, and A ? Q(A) is an arithmetical ring if and only if A is a semihereditary ring.