1.

On JBRings





Huanyin CHEN Department of Mathematics Hunan Normal University Changsha 410006 China.《数学年刊B辑(英文版)》,2007年第28卷第6期


A ring R is a QBring provided that aR bR=R with a,b∈R implies that there exists a y∈R such that a by∈R_q~(1).It is said that a ring R is a JBring provided that R/J(R)is a QBring,where J(R)is the Jacobson radical of R.In this paper,various necessary and sufficient conditions,under which a ring is a JBring,are established.It is proved that JBrings can be characterized by pseudosimilarity.Furthermore,the author proves that R is a JBring iff so is R/J(R)~2.

2.

On Weakly Semicommutative Rings





CHEN WEIXING CUI SHUYING《东北数学》,2011年第2期


A ring R is said to be weakly semicommutative if for any a,b∈R, ab=0 implies aRb（？）Nil（R）,where Nil（R） is the set of all nilpotent elements in R. In this note,we clarify the relationship between weakly semicommutative rings and NIrings by proving that the notion of a weakly semicommutative ring is a proper generalization of NIrings.We say that a ring R is weakly 2primal if the set of nilpotent elements in R coincides with its Levitzki radical,and prove that if R is a weakly 2primal ring which satisfiesαcondition for an endomorphismαof R（that is,ab=0（？）aα（b）=0 where a,b∈R） then the skew polynomial ring R[x;α] is a weakly 2primal ring,and that if R is a ring and I is an ideal of R such that I and R/I are both weakly semicommutative then R is weakly semicommutative. Those extend the main results of Liang et al.2007（Taiwanese J.Math.,11（5）（2007）, 13591368） considerably.Moreover,several new results about weakly semicommutative rings and NIrings are included.

3.

分次单环的结构





朱彬《东北数学》,2003年第19卷第3期


A characterization of grsimple rings is given by using the notion of componentwisedense subrings of a full matrix ring over a division ring. As a consequence, any Ggraded full matrix ring over a division ring is isomorphic to a dense subring of a full matrix ring with a good Ggrading. Some conditions for a grading of a full matrix ring to be isomorphic to a good one are given, which generalize some results in: Dascascu, S., Lon, B., Nastasescu, C. and Montes, J. R., Group gradings on full matrix rings, J. Algebra, 220(1999), 709728.

4.

Generalized Differential Identities of （Semi）Prime Rings





Feng WEI《数学学报(英文版)》,2005年第21卷第4期


Let R be a semiprime ring with characteristic p≥0 and RF be its left Martindale quotient ring. If ф（Xi^△j） is a reduced generalized differential identity for an essential ideal of R, then ф（Zije（△j ）） is a generalized polynomial identity for RF, where e（△j） are idempotents in the extended centroid of R determined by △j. Let R be a prime ring and Q be its symmetric Martindale quotient ring. If ф（Xi△j） is a reduced generalized differential identity for a noncommutative Lie ideal of R, then ф（Zij） is a generalized polynomial identity for [R, R]. Moreover, if ф（Xi△j） is a reduced generalized differential identity, with coefficients in Q, for a large right ideal of R, then ф（Zij） is a generalized polynomial identity for Q.

5.

The K_2groups over Finaite Commutative Rings





南基洙 田子德《东北数学》,2002年第2期


The present note determines the structure of the K2group and of its subgroup over a finite commutative ring R by considering relations between R andfinite commutative local ring Ri (1 < i < m), where R Ri and K2(R) =K2(Ri). We show that if charKi= p (Ki denotes the residual field of Ri), then K2(Ri) and its subgroups must be pgroups.

6.

The K2groups over Fimite Commutative Rings





南基洙 田子德《东北数学》,2002年第18卷第2期


The present note determines the structure of the K2group and of itssubgroup over a finite commutative ring R by considering relations between R andfinite commutative local ring Ri (1 ≤ i ≤ m), where R ≌ m i=1 Ri and K2(R) ≌ m i=1 K2(Ri). We show that if charKi = p (Ki denotes the residual field of Ri), then K2(Ri) and its subgroups must be pgroups.

7.

ON EULER CHARACTERISTIC OF MODULES~(**)





佟文廷《数学年刊B辑(英文版)》,1989年第1期


This paper gives a characteristic property of the Euler characteristic for IBN rings. The following results: are proved. (1) If R is a commutative ring, M, N are two stable free Rmodules, then χ(MN)=χ(M)χ(N), where χ denotes the Euler characteristic. (2) If f: K_0(R)→Z is a ring isomorphism, where K_0(R) denotes the Grothendieck group of R, K_0(R) is a ring when R is commutative, then f([M])=χ(M) and χ(MN)=χ(M)χ(N) when M, N are finitely generated projective Rmodules, where.the isomorphism class [M] is a generator of K_0(R). In addition, some applications of the results above are also obtained.

8.

主理想环上子群Gr在线性群中的扩群





卫宗礼 曲贺梅《数学季刊》,2008年第23卷第4期


Suppose R is a principal ideal ring,R~* is a multiplicative group which is composed of all reversible elements in R,and M_n(R),GL(n,R),SL(n,R) are denoted by, M_n(R)={A=(a_(ij))_(n×n)a_(ij)∈R,i,j=1,2,…,n},GL(n,R) = {gg∈M_n(R),detg∈R~*},SL(n,R) = {g∈GL(n,R)det g=1},SL(n,R)≤G≤GL(n,R)(n≥3),respectively, then basing on these facts,this paper mainly focus on discussing all extended groups of G_r={(AB OD)∈GA∈GL(r,R),(1≤r

9.

The structure of a class of Zlocal rings





WU Tongsuo & LU Dancheng Department of Mathematics Shanghai Jiao Tong University Shanghai 200240 China Department of Mathematics Suzhou University Suzhou 215006 China《中国科学A辑(英文版)》,2006年第49卷第10期


A local ring R is called Zlocal if J(R) = Z(R) and J(R)2 = 0. In this paper the structure of a class of Zlocal rings is determined.

10.

Jclean and Strongly Jclean Rings





XIANG YUEMING OUYANG LUNQUN《数学研究通讯：英文版》,2018年第3期


Let R be a ring and J(R) the Jacobson radical. An element a of R is called(strongly) Jclean if there is an idempotent e ∈ R and w ∈ J(R) such that a = e + w(and ew = we). The ring R is called a(strongly) Jclean ring provided that every one of its elements is(strongly) Jclean. We discuss, in the present paper,some properties of Jclean rings and strongly Jclean rings. Moreover, we investigate Jcleanness and strongly Jcleanness of generalized matrix rings. Some known results are also extended.

11.

斜Armendariz矩阵环





杨刚 刘仲奎 王彦军《数学研究与评论》,2010年第30卷第6期


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

12.

Noetherian Skew Group Rings of F.C Groups





HU Changliu WANG Jianping 《数学季刊》,2006年第21卷第3期


Let R *θG be the skew group ring with a F.C group G and the group homomrphismθfrom G to Aut(R), the group of automorphisms of the ring R. In this paper,the necessary and sufficient condition such that R*θG will be Noetherian is given, which generalizes the results of I.G. connel.

13.

Semi Group Rings Which are Chinese Ring





W.B.Vasantha Kandasamy《数学研究与评论》,1993年第3期


In this paper we obtain conditions under which a semigroup ring is a Chinese ring.Further we define what are called weakly Chinese rings and study them.The authors in[1]called a commutative ring R to be a Chinese ring if,given elements a,b∈R and idealI,J(?)R such that a≡b(I J)there exists an element c∈R such that c≡a(I)andc≡b(J).For more properties about Chinese rings please refer[1].

14.

一个四元数矩阵方程的可解性 被引次数：3





曹文胜《高校应用数学学报(英文版)》,2002年第17卷第4期


§ 1 IntroductionL et R be the real number field,C=R Ri be the complex numberfield,and H=C Cj=R Ri Rj Rk be the quaternion division ring over R,where k:=ij= ji,i2 =j2 =k2 = 1 .Ifα=a1 +a2 i+a3 j+a4 k∈ H ,where ai∈ R,then letα=a1  a2 i a3 j a4 k bethe conjugate ofα.L et Hm× nbe the setof all m× n matrices over H.If A=(aij)∈ Hn× n ,L etATbe the transpose matrix of A,A be the conjugate matrix of A,and A* =(aij) T be thetranspose conjugate matrix of A.A∈Hn× nis said…

15.

Existence Results for Nonlinear Subelliptic Equations on the Heisenberg Group





罗学波 张吉慧《东北数学》,1999年第4期


§ 1.Introduction The aim of this paper is to establish existence results,by monotone method,for theproblemΔHnu + f((z,t) ,u) =0 in D,u D =0 (1 .1 )where D is an open subset of the Heisenberg group HnandΔHn is the subelliptic Laplacian on Hn.We recall that Hnis the Lie group whose underlying manifold is Cn× R,n∈N,endowed with the group law(z,t) (z′,t′) =(z + z′,t+ t′+ 2 Imz .z′) ,(1 .2 )where for z,z′∈ Cnwe have letz .z′= nj=1zjz′j.Set zj=xj+ iyj.Then (x1 ,… ,xn…

16.

K─f环的张量积





周伟《数学研究与评论》,1994年第14卷第1期


A multiplication is introduced into the tensor products of K一lattice ordered modules、where K is a commutative lottice ordered ring with identity.It is shown that the positivecone of the Abelian lgroup of the tensor products is closed under this multiplication andthat the tensor products of Kf rings is a Klattice ordered ring.

17.

Maximal Quotient Rings of Endomorphisms of Quasigenerators





朱胜林《数学研究与评论》,1989年第2期


O.Preliminaries. Let R be an associative ring with identity, and let ModR denote the category of all unital right Rmodules. A set of right ideal of R is called a Gabriel topology on R if satisfies T1. If I∈ and I J, then J∈. T2. If I and J belong to, then I∩J∈. T3. I∈ and r∈R, then (I:r)={x∈R:rx∈I}∈. T4. If I is a right ideal of R and there exists J∈ such that (I:r)∈ for every r∈J, then I∈.

18.

COINDUCED REPRESENTATIONS AND INJECTIVE MODULES FOR HYPERALGEBRA b_r





王建磐《数学年刊B辑(英文版)》,1983年第3期


Let G be a simply connected semisimple linear algebraic group over an algebraicallyclosed field of positive characteristic p,B its Borel subgroup,and b_r the rth standard subalgebra of the hyperalgebra of B.Assume the roots in B to be negative.Using the coinduced representations,in this paper the author proves:(1)J(r,λ)=St_r((p~r1)δ+λ)is the b_rinjective envelope of the onedimensionalb_rmodule λ,where St_r is the rth Steinberg module of G,and δ half the sum of the positiveroots.(2)With respect to the natural homomorphism p_(rs):J(r,λ)→J(s,λ)(r≤s),J(∞,λ)=lim J(r,λ)is the Binjective envelope of Bmodule λ.The above conclusions positively answer two questions posed by J.E.Humphreys atShanghai in 1980.Moreover,this paper gives a complete description of injective b_rmodules.

19.

弱对偶环





魏俊潮 孙建华《东北数学》,2004年第20卷第4期


In This paper, the concept of weakly dual ring is introduced, which is a proper generalization of the dual ring. If R is a right weakly dual ring, then (1) Z(RR) = J(R); (2) If R is also a zerodivision power ring, then R is a right APinjective ring. In addition, some properties of weakly dual rings are given.

20.

On Skew Triangular Matrix Rings





《数学研究通讯：英文版》,2016年第3期


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

