Γ-环的诸根的关系及Kyuno的一个结果

Let M={a, b, c,…} and Γ={α,β,γ,…} be additive abefian groups. If for all a, b, C∈M and all a, β∈Γ, the following conditions are satisfied: (0) aab∈M, (1) (a b)ac=aac bar, a(α β)b=aab aβb, aa(b e)=aab aac, (2) (aab)[βc=aa(bβc),then M is called a Γ-ring. If for all a, b, ,∈M ahd all tx, β∈Γ, the following conditions are satisfied:     

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.     

A Note on Pseudo Ideals of Fields

Definition A subring A of a ring R is called a pseudo ideal, if for any r∈R and α∈A, r~2α∈A and αr~2∈A. The subring (R)~2={Σ±r_1~2…r_i~2|r_i∈R, t∈N} is called the square closed ring of R. Lemma The square closed ring of a division ring D is the intersection of all nontrivial pseudo ideals of D. Theorem Ⅰ If a simple ring R with a unit has a nontrivial pseudo ideal A containing an invertible element, R has characteristic p=2. In particular, the necessary condition for a division ring containing a nontrivial pseudo ideal is     

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     

Maximal Quotient Rings of Endomorphisms of Quasigenerators

O.Preliminaries. Let R be an associative ring with identity, and let Mod-R denote the category of all unital right R-modules. 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∈.     

On π-regularity of General Rings

A general ring means an associative ring with or without identity.An idempotent e in a general ring I is called left （right） semicentral if for every x ∈ I,xe=exe （ex=exe）.And I is called semiabelian if every idempotent in I is left or right semicentral.It is proved that a semiabelian general ring I is π-regular if and only if the set N （I） of nilpotent elements in I is an ideal of I and I /N （I） is regular.It follows that if I is a semiabelian general ring and K is an ideal of I,then I is π-regular if and only if both K and I /K are π-regular.Based on this we prove that every semiabelian GVNL-ring is an SGVNL-ring.These generalize several known results on the relevant subject.Furthermore we give a characterization of a semiabelian GVNL-ring.     

关于拟AP内射模的一些研究

Let R be a ring. R is called right AP-injective if, for any a ∈ R, there exists a left ideal of R such that lr(a) = Ra (?) Xa. We extend this notion to modules. A right .R-module M with 5 = End(MR) is called quasi AP-injective if, for any s ∈ S, there exists a left ideal Xs of S such that ls(Ker(s)) = Ss (?) Xs. In this paper, we give some characterizations and properties of quasi AP-injective modules which generalize results of Page and Zhou.     

具有对称自同态的环及其扩张

Let R be a ring with an endomorphism α and an α-derivation δ. We introduce the notions of symmetric α-rings and weak symmetric α-rings which are generalizations of symmetric rings and weak symmetric rings, respectively, discuss the relations between symmetricα-rings and related rings and investigate their extensions. We prove that if R is a reduced ring and α(1) = 1, then R is a symmetric α-ring if and only if R[x]/(x n) is a symmetric ˉα-ring for any positive integer n. Moreover, it is proven that if R is a right Ore ring, α an automorphism of R and Q(R) the classical right quotient ring of R, then R is a symmetric α-ring if and only if Q(R) is a symmetric ˉα-ring. Among others we also show that if a ring R is weakly 2-primal and(α, δ)-compatible, then R is a weak symmetric α-ring if and only if the Ore extension R[x; α, δ] of R is a weak symmetric ˉα-ring.     

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.     

On Weakly Semicommutative Rings

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 NI-rings by proving that the notion of a weakly semicommutative ring is a proper generalization of NI-rings.We say that a ring R is weakly 2-primal if the set of nilpotent elements in R coincides with its Levitzki radical,and prove that if R is a weakly 2-primal ring which satisfiesα-condition 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 2-primal 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）, 1359-1368） considerably.Moreover,several new results about weakly semicommutative rings and NI-rings are included.