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.     

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.     

质环的求导和交换性

In this paper, we generalize some corresponding results of [1-4]. We obtain the main results as the following:Theorem 1 Let R be a prime ring of characteristic not 2 with nontrivial derivations d₁,d₂ and let U be a nonzero ideal of R . If C is the center of R, then the following conditions are equivalent : (i)d₁,d₂(x)∈C for all x∈U; (ii) [d₁(x),d₂(y)]∈C for all x,y∈U; (iii) d₁(x)d₂(y)+d₂(x)d₁(y)∈C for all x,y∈U; (iv) R is commutative.Theorem 2 Let R be a prime ring with nontrivial derivations d₁,d₂,…, d_n and U be a nonzero ideal of R. Let C be the center of R. If d₁(x₁)d₂(x₂)…d_n(x_n)∈C for all x₁, x₂…x_n)∈U, then R is commutative.     

TOPOLOGICAL CHARACTERIZATIONS OF THE EXTENDING PROPERTY OF RINGS

A commutative ring R is called extending if every ideal is essential in a direct summand of RR. The following results are proved： （1） C（X） is an extending ring if and only if X is extremely disconnected; （2） Spec（R） is extremely disconnected and R is semiprime if and only if R is a nonsingular extending ring; （3） Spec（R） is extremely disconnected if and only if R/N（R） is an extending ring, where N（R） consists of all nilpotent elements of R. As an application, it is also shown that any Gelfand nonsingular extending ring is clean.     

半质环的两个交换性结果

定理1 设R是半质环,m,n是固定正整数,且n>1.如果R满足条件(x^my)ⁿ-x^my∈Z(R),?x,y∈R,则R是交换环.定理2 设R是半质环,m,n,s,t是固定正整数,且(m+n)t=s+1,mt>1.如果R满足条件[x^m,yⁿ]^t-[x,y^s]∈Z(R),?x,y∈R,则R是交换环.     

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].     

TWO COMMUTATIVITY RESULTS FOR SEMI-PRIME RINGS

1.Introduction.In 1980.V.Gupta [2] provedTheorem A Let R be a semi-prime ring with unit satisfying(i)[x~n,y]-[x,y~n]∈Z(R) (ii)[x~(n+1),y]-[x,y~(n+1)]∈Z(R)for all x,y∈R and a fixed integer n>1,then R is commutative.This theoremimproved a theorem which was established by Harmanci [4] in 1977 that if aring R with unit satisfies the identity(i)[x~n,y]=[x,y~n] (ii)[x~(n+1),y]=[x,y~(n+1)] for all x,y∈Rand a fixed integer n>1,then R is commutative.Later,in 1982,Guo Yuan     

On Regular Power-Substitution

The necessary and sufficient conditions under which a ring satisfies regular power-substitution are investigated. It is shown that a ring R satisfies regular powersubstitution if and only if a-b in R implies that there exist n ∈ N and a U E GLn （R） such that aU = Ub if and only if for any regular x ∈ R there exist m,n ∈ N and U ∈ GLn（R） such that x^mIn = xmUx^m, where a-b means that there exists x,y, z∈ R such that a =ybx, b = xaz and x= xyx = xzx. It is proved that every directly finite simple ring satisfies regular power-substitution. Some applications for stably free R-modules are also obtained.     

实赋范数性空间内的Lp-正交

In this paper we have proved the theorem: Let p>1, and E be a real normed linear space, if the L^p-orthogonality in E satisfies one of the following conditions 1) homogeneity 2) additivity 3) x⊥_L^py implies x⊥_Jy 4) x⊥_Jy implies x⊥_L^py, then E is an abstract Euclidean space and there must be p=2. We also proved anR. C. James' result-If for every element x of a normed linear space E therecan be found a nonzero element orthogonal to x by Roberts' definition in each two dimensional linear subset containing x, then E is an abstract Euclidean space——which had not been proved. Finally, we point out that if one of the James′,L^p-,isosceles, Pythagorean and (α,β)-orthogonalities defined in E implies Roberts′ orthogonality then E is an abstract Euclidean space.     

PF环与群环的Grothendieck群

Let R be a commutative ring with 1, G an Abelian group, RG the group ring on R and G. In this paper we gave some properties of PF- rings in which f. g. projective modules are free. The Grothendieck groups K₀(RG) for some cases are given. In addition, for the ring R with the unimodular column property, we proved the following result: K₀(RG) ≈K₀(R), hence if R ∈PF, then K₀(RG)≈Z .