The K_2-groups over Finaite Commutative Rings

The present note determines the structure of the K2-group and of its subgroup over a finite commutative ring R by considering relations between R and finite 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 p-groups.     

交换环上上三角矩阵保对合与保立方幂等的模自同构

Suppose R is a commutative ring with 1, and 2 is a unit of R. Let Tn(R) be the n × n upper triangular matrix modular over R, and let (?)i(R) (i=2 or 3) be the set of all R-module automorphisms on Tn(R) that preserve involutory or tripotent. The main result in this paper is that f ∈ (?)i(R) if and only if there exists an invertible matrix U ∈ Tn(R) and orthogonal idempotent elements e1,e2,e3 ande4 in R with such that where     

ON THE QUOTIENT RING OF COMMUTATIVE RINGS WITH ACC^{\perp}ON ANNIHILATOR IDEALS.

The author concludes that every commutative ring with ascending chain condition an annihilator ideals has a Kasch quotient ring, which generalizes the Theorem^{[1]} that every commutative noetherian ring has a Kasch quosient ring. If follows that if R is a commutative ting with acc^{\perp}，then that Q(R) is semiprimary is equivalent to that it is perfect, or to that R satisfies regular condition. Besides, that Q(R) is quasi-frobenius equals that Q(R) is FPF or PF, and that Q(R) is artinian equals that R/N, are of finite dimension, i=1,2,\ldots,n. N_{i}=J^{i}\cap R.     

The automorphism group of a generalized extraspecial p-group

In this paper, the automorphism group of a generalized extraspecial p-group G is determined, where p is a prime number. Assume that |G| = p 2n+m and |ζG| = p m , where n 1 and m 2. (1) When p is odd, let Aut G G = {α∈ AutG | α acts trivially on G }. Then Aut G G⊿AutG and AutG/Aut G G≌Z p-1 . Furthermore, (i) If G is of exponent p m , then Aut G G/InnG≌Sp(2n, p) × Z p m-1 . (ii) If G is of exponent p m+1 , then Aut G G/InnG≌ (K Sp(2n-2, p))×Z p m-1 , where K is an extraspecial p-group of order p 2n-1 . In particular, Aut G G/InnG≌ Z p × Z p m-1 when n = 1. (2) When p = 2, then, (i) If G is of exponent 2 m , then AutG≌ Sp(2n, 2) × Z 2 × Z 2 m-2 . In particular, when n = 1, |AutG| = 3 · 2 m+2 . None of the Sylow subgroups of AutG is normal, and each of the Sylow 2-subgroups of AutG is isomorphic to H K, where H = Z 2 × Z 2 × Z 2 × Z 2 m-2 , K = Z 2 . (ii) If G is of exponent 2 m+1 , then AutG≌ (I Sp(2n-2, 2)) × Z 2 × Z 2 m-2 , where I is an elementary abelian 2-group of order 2 2n-1 . In particular, when n = 1, |AutG| = 2 m+2 and AutG≌ H K, where H = Z 2 × Z 2 × Z 2 m-1 , K = Z 2 .     

Linear Group over a Class of Ring R

Let R be a commutative ring^[1] with identical element 1 and maximal ideal M_i where i\in N and N is an ordered indieatrix set. Let the mapping $f:R\rightarrow \prod\limits_{i \in N} {R/{M_i}}$, be a ring homomorphism from R onto $[\prod\limits_{i \in N} {R/{M_i}} \]$, where $[\prod\limits_{i \in N} {R/{M_i}} \]$ is the direct product of residual fields B/M_i. In this paper, it is proved that if A \in GL_n(R), then A=BH_1……H_k-1, where res B=1 and H_1,\cdots, H_k-1 are the symmetries. Furthermore, the bound of the positive integer number K is investigated. In particular, the author gives the smallest number l(A) of symmetric factors in the products which expresses the elements of G_n= {A\in GLn(R)| det A=±l}. Consequently, the l(A) problems discussed in [2, 3, 4] are special cases of this paper.     

半质环交换性的一点注记

Let R be a semi-prime ring, C be the center of R. Let F_i (x, y) (i = 1, 2) be a product of the m times x's and n times y's.In this paper following theorem is proved: (I ) implies (Ⅱ), where( Ⅰ )If f₁(x,y) -f₂(x,y) ∈C for every x,y in R, then R is commutative;(Ⅱ)If f₁ (x,y) + f₂(x, y) ∈C for every x,y in R, then R is commutative.Thus very short proves of some theorems of references[5], [8], [9] are be given.     

The Tensor Products of Nilrings with Bounded Index of Nilpotence

R is a ring, for any nilpotent element r∈R if there exists a fixed integer nsuch that r~n=0, then R is said to be with bounded index of nilpotence, theleast of such integer n is called the index of R, denoted by i(R). If R is a nil ring with bounded index i(R)=n, R′ is a commutative ring,A·A·Klein〔1〕 has discussed the property of bounded index of nilpotence of RR′. In this paper we shall discuss the properties of bounhed index of nil-potence of RR′ when R′ is not commutative.     

ON THE QUOTIENT RING OF COMMUTATIVE RINGS WITH ACC⊥ON ANNIHILATOR IDEALS

The author concludes that every commutative ring with ascending chain conditionan annihilator ideals has a Kasch quotient ring,which generalizes the Theorem thatevery commutative noetherian ring has a Kasch quosient ring.If follows that if R is acommutative ring with acc~⊥,then that Q(R)is semiprimary is equivalent to that it isperfect,or to that R satisfies regular condition.Besides,that Q(R)is quasi-frobeniusequals that Q(R)is FPF or PF,and that Q(R)is artinian equals that B/N,are of finitedimension,i=1,2,…,n.N_i,=J~i∩R.     

一个四元数矩阵方程的可解性

§ 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…     

具有单位元素1之非结合环的交换性

In thin paper, we prove that a nonassociative ring R with 1 satisfying identity(?) if R is w-torsion-free, where ai is an integer (i =1, 2,…, t) and the so-called torsion characteristic number W≠0 is a definite positive integer depending on R, then R is commutative.     

主理想环上子群Gr在线性群中的扩群

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) = {g|g∈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)∈G|A∈GL(r,R),(1≤r相似文献   

一类替换环的偏序$K_{0}$-群的无扭性

A ring R is called orthogonal if for any two idempotents e and f in R, the condition that e and f are orthogonal in R implies the condition that [eR] and [fR] are orthogonal in K0（R）^＋, i.e., [eR]∧[fR] = 0. In this paper, we shall prove that the K0-group of every orthogonal, IBN2 exchange ring is always torsion-free, which generalizes the main result in [3].     

ON EULER CHARACTERISTIC OF MODULES~(**)

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 R-modules, 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 R-modules, where.the isomorphism class [M] is a generator of K_0(R). In addition, some applications of the results above are also obtained.     

OSCILLATION CRITERIA FOR A CLASS OF HIGHER ORDER NEUTRAL DIFFERENTIAL EQUATIONS

We establish a new oscillation condition for even order neutral type differential equation of the following formwhere f is continuous and exists 1 ≤ k ≤ m(?)(t) ≡ Tk(t), a, (?)i ∈ C([0, ∞], R), i = 1, …,m. such that 0 ≤ a(t) ≤ L and     

Co-commuting Mappings of Generalized Matrix Algebras

Let G be a generalized matrix algebra over a commutative ring R and Z(G)be the center of G.Suppose that F,T:G→G are two co-commuting R-linear mappings,i.e.,F(x)x=xT(x) for all x∈G.In this note,we study the question of when co-commuting mappings on G are proper.     

Real submanifolds in complex spaces

Let(z_(11),..., z_(1N),..., z_(m1),..., z_(mN), w_(11),..., w_(mm)) be the coordinates in C~(mN) +m~2. In this note we prove the analogue of the Theorem of Moser in the case of the real-analytic submanifold M defined as follows W = ZZ~t+ O(3),where W = {w_(ij)}_(1≤i,j≤m)and Z = {z_(ij) }_(1≤i≤m, 1≤j≤N). We prove that M is biholomorphically equivalent to the model W = ZZ~t if and only if is formally equivalent to it.     

MEROMORPHIC SOLUTIONS OF A TYPE OF SYSTEM OF COMPLEX DIFFERENTIAL-DIFFERENCE EQUATIONS

Applying Nevanlinna theory of the value distribution of meromorphic functions,we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference equations of the following form∑nj=1aj(z)f1(λj1)(z+cj) = R2(z, f2(z)),∑nj=1βj(z)f2(λj2)(z+cj)=R1(Z,F1(z)).(*)where λij(j = 1, 2, ···, n; i = 1, 2) are finite non-negative integers, and cj(j = 1, 2, ···, n)are distinct, nonzero complex numbers, αj(z), βj(z)(j = 1, 2, ···, n) are small functions relative to fi(z)(i = 1, 2) respectively, Ri(z, f(z))(i = 1, 2) are rational in fi(z)(i = 1, 2)with coefficients which are small functions of fi(z)(i = 1, 2) respectively.     

ARTINIAN RADICAL AND ITS APPLICATION

Let R be a left and right Neotherian ring with identity. Let A be the Artinian radical.Lenagan [3]pointed out that R has Artinian quotient ring if A=O and the Krull dimensionof R is one. In this paper first the structure of Artinian radical is investigated. Then forR with Krull dimension one the author gives a necessary and sufficient condition under whichR has Artinian quotient ring. The main results are as follows: (i) A=eR, where e is acentral idempotent element of R, if and only if r(A)~λ=l(A)~λ=(∩ i=1 ,…, n_k k=1, …, p(a_~(k)))~λ, where λ isa positive integer, p(a_i~(k)) are prime ideals of i~ and ~(A)(lA)) is the notation of right(:eft)annihilatorof A(see Theorem 7). (ii) In the case(i)R=A~η(A)~λ. (iii) If R has Krulldimension one, then R has Artinian quotient ring if and only if there exists a positivointeger λ such that (A)~λ=l(A)~λ=(∩ i=1 ,…, n_k k=1, …, p(a_~(k)))~λ.     

SOME NECESSARY AND SUFFICIENT CONDITIONS FOR EXISTENCE OF POSITIVE SOLUTIONS FOR THIRD ORDER SINGULAR SUBLINEAR MULTI-POINT BOUNDARY VALUE PROBLEMS

We mainly study the existence of positive solutions for the following third order singular multi-point boundary value problem{x(3)(t) + f(t, x(t), x′(t)) = 0, 0 t 1,x(0)-m1∑i=1 αi x(ξi) = 0, x′(0)-m2∑i=1 βi x′(ηi) = 0, x′(1)=0,where 0 ≤ ai≤m1∑i=1 αi 1, i = 1, 2, ···, m1, 0 ξ1 ξ2 ··· ξm1 1, 0 ≤βj≤m2∑i=1βi1,J=1,2, ···, m2, 0 η1 η2 ··· ηm2 1. And we obtain some necessa βi =11, j = 1,ry and sufficient conditions for the existence of C1[0, 1] and C2[0, 1] positive solutions by constructing lower and upper solutions and by using the comparison theorem. Our nonlinearity f(t, x, y)may be singular at x, y, t = 0 and/or t = 1.