关于拟AP内射模的注记

Let R be a ring.A right R-module M with S=End(MR)is called aquasi AP-injective module,if,for any s ∈ S,there exists a left ideal X_s of S such thatl_s(ker s)=SsX_s.Let M be a quasi AP-injective module which is a self-generator.We show that for such a module,if S is semiprime,then every maximal kernel of S isa direct summand of M.Furthermore,if ker(a_1)ker(a_2a_1)ker(a_3a_2a_1)satisfy the ascending conditions for any sequence a_1,a_2,a_3,...∈ S,then S is rightperfect.In this paper,we give a series of results which extend and generalize resultson AP-injective rings.     

正则环和GP-内射环

A ring R is called left Gp-injective if for any a∈R, there exists a positive integer n such that any left R-homomorphism of Raⁿ into R extends to one of R into R. In this paper, we prove that the centre of semiprime (left nonsingular) left GP- injective ring is regular ring, and improve some propositions in [3].     

T正则环与TV环

In this paper, we generalize flat modules, regular rings and V- rings to the situation of a hereditary torsion theory, that a ring R is regular if and only if R is a left nonsingular ring and for every essential left ideal o of R, and for eve ry element a∈o there exists a element a′∈o , such that a= aa′. (theorem 3.3).     

On Maximal Injectivity

A right R-module E over a ring R is said to be maximally injective in case for any maximal right ideal m of R, every R-homomorphism f ： m → E can be extended to an R-homomorphism f^1 ： R → E. In this paper, we first construct an example to show that maximal injectivity is a proper generalization of injectivity. Then we prove that any right R-module over a left perfect ring R is maximally injective if and only if it is injective. We also give a partial affirmative answer to Faith＇s conjecture by further investigating the property of maximally injective rings. Finally, we get an approximation to Faith＇s conjecture, which asserts that every injective right R-module over any left perfect right self-injective ring R is the injective hull of a projective submodule.     

$GP$-$V^{''}$-环的非奇异性与正则性

In this paper,we introduce a non-trivial generalization of ZI-rings-quasi ZI-rings.A ring R is called a quasi ZI-ring,if for any non-zero elements a,b ∈ R,ab = 0 implies that there exists a positive integer n such that an = 0 and anRbn = 0.The non-singularity and regularity of quasi ZI,GP-Vˊ-rings are studied.Some new characterizations of strong regular rings are obtained.These effectively extend some known results.     

The Non-Singularity and Regularity of GP-Vˊ-Rings

In this paper,we introduce a non-trivial generalization of ZI-rings-quasi ZI-rings.A ring R is called a quasi ZI-ring,if for any non-zero elements a,b ∈ R,ab = 0 implies that there exists a positive integer n such that an = 0 and anRbn = 0.The non-singularity and regularity of quasi ZI,GP-Vˊ-rings are studied.Some new characterizations of strong regular rings are obtained.These effectively extend some known results.     

RESEARCH ANNOUNCEMENTS A Generalization if Goldie's Theorem

In this paper we proved two theorems which are generalizations of Goldie’sTheorem, Definition 1.A ring R is said to be a right G-ring,if R satisfies (i)For any nonezero left ideal L of R,there exists 0≠x∈L,for any S,t∈R, st≠0,xt≠0 imply sxt≠0. (ii)For any x∈R,the right Goldie’s dimension of xR is finite. Definition 2.Let Δ be a division ring and N be a veetor space over Δ,a     

Morphic环的扩张

A ring R is called left morphic, if for any a ∈ R, there exists b ∈ R such that 1R(a) = Rb and 1R(b) = Ra. In this paper, we use the method which is different from that of Lee and Zhou to investigate when R[x, σ]/(xn) is (left) morphic and when the ideal extension E(R, V) is (left) morphic. It is mainly shown that: (1) If σis an automorphism of a division ring R, then S = R[x,σ]/(xn) (n ＞ 1) is a special ring. (2) If d, m are positive integers and n = dm, then E(/n, mZn) is a morphic ring if and only if gcd(d, m) = 1.     

The Structure of Almost Quasi-Regular Rings

Let R be a ring. It is well-known that R is a semigroup with respect to the composition a b=a b-ab for a,b∈R. An element a of R is said to be (right; left) quasi-regular if there exists an element b in R such that (a b=0;     

Dependent sets of a family of relations of full measure on a probability space

For a probability space (X, B,μ) a subfamily F of theσ-algebra B is said to be a regular base if every B∈B can be arbitrarily approached by some member of F which contains B in the sense of the measure theory. Assume that {γr}γ∈Γis a countable family of relations of the full measure on a probability space (X,B,μ), i.e. for everyγ∈Γthere is a positive integer sγsuch that Rγ(?)Xsγwithμsγ(Rγ) = 1. In the present paper we show that if (X, B,μ) has a regular base, the cardinality of which is not greater than the cardinality of the continuum, then there exists a set K(?)X withμ*(K) = 1 such that (x1,...,xsγ)∈γr for anyγ∈Γand for any sγdistinct elements x1,..., xsγof K, whereμ* is the outer measure induced by the measureμ. Moreover, an application of the result mentioned above is given to the dynamical systems determined by the iterates of measure-preserving transformations.     

关于AP-内射环的一个注记

本文的主要目的是讨论AP-内射环中的两个问题:(1)环R是正则的当且仅当R是左AP-内射的左PP-环;(2)如果R是左AP-内射环,那么R是内射环当且仅当R是弱内射环.因此我们推广了内射环的一些结果,与此同时我们还取得了一些新的结果.     

关于AP-内射环和AGP-内射环(英文)

本文研究了AP-内射环和AGP-内射环的von Neumann正则性问题.利用P-内射环和Y J-内射环的研究方法及GW-理想的性质,得到了AP-内射环和AGP-内射环是(强)正则环的一些条件.推广了文献[7,12,14]中的相关结果.     

Group homomorphisms inducing isomorphisms in cohomology

In this article, we call a ring R right generalized semiregular if for any a ∈ R there exist two left ideals P, L of R such that lr(a) = P⊕ L, where P ? Ra and Ra ∩ L is small in R. The class of generalized semiregular rings contains all semiregular rings and all AP-injective rings. Some properties of these rings are studied and some results about semiregular rings and AP-injective rings are extended. In addition, we call a ring R semi-π-regular if for any a ∈ R there exist a positive integer n and e ² = e ∈ a ⁿ R such that (1 ? e)a ⁿ  ∈ J(R), the Jacobson radical of R. It is shown that a ring R is semi-π-regular if and only if R/J(R) is π-regular and idempotents can be lifted modulo J(R).     

关于Jacobson根的一些研究

主要证明了:(i)假设R是右广义半正则右ACS-环,若J(R)∩I=J(I)对于R的任意右理想I都成立,则J(R)=Z(RR);(ii)如果R是右AP-内射环且R的每个奇异单右R-模是GP-内射,则对于R的任意右理想I都有J(R)∩I=J(I).     

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.     

RECENT PROGRESS ON SPHERICAL HARMONIC APPROXIMATION MADE BY BNU RESEARCH GROUP ——In memory of Professor Sun Yongsheng

As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years （2001-2005）. The main topics are： convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.     

A unified approach to certain problems of best local approximation

In this paper we study best local quasi-rational approximation and best local approximation from finite dimensional subspaces of vectorial functions of several variables. Our approach extends and unifies several problems concerning best local multi-point approximation in different norms.     

Boundedness for commutators of multipliers on Hardy type spaces

In this paper, we study the commutators generalized by multipliers and a BMO function. Under some assumptions, we establish its boundedness properties from certain atomic Hardy space Hb^p（R^n） into the Lebesgue space L^p with p 〈 1.