Supersemiprime Rings and Supersemiprime Radical

In this paper,supersemiprime ring is introduced,relations among the supersemiprime rings and the some rings around it are discussed,supersemiprime radical and its properties are studied.

The Linear Combination of Derivations in Prime Rings

In this paper we discuss the linear combination of derivations in prime rings which was initiated by Niu Fengwen. Our aim is to improve Niu's result. For the proof of the main theorem we generalize a result of Bresar and obtain a lemma that is of independent interest.

Cocentralizing Generalized Derivations in Prime Rings

Let R be a prime ring with center Z and S包含 R. Two mappings D and G of R into itself are called cocentralizing on S if D（x）x - xG（x） ∈ Z for all x ∈S. The main purpose of this paper is to describe the structure of generalized derivations which are cocentralizing on ideals, left ideals and Lie ideals of a prime ring, respectively. The semiprime ease is also considered.

Generalized Derivations in Prime Rings

Let R be a ring, a ,b ∈ R, ( D , α ) and (G , β ) be two generalized derivations of R . It is proved that if aD ( x ) = G ( x )b for all x ∈ R, then one of the following possibilities holds: (i) If either a or b is contained in C , then α = β= 0 and there exist p , q ∈ Qr ( RC) such that D ( x )= px and G ( x )= qx for all x ∈ R;(ii) If both a and b are contained in C , then either a = b= 0 or D and G are C-linearly dependent;(iii) If neither a nor b is contained in C , then there exist p , q ∈ Qr ( RC) and w ∈ Qr ( R) such that α ( x ) = [ q ,x] and β ( x ) = [ x ,p] for all x ∈ R, whence D ( x )= wx-xq and G ( x )= xp + avx with v ∈ C and aw-pb= 0.

Notes on Automorphisms of Prime Rings

Let R be a prime ring, L a noncentral Lie ideal and a nontrivialautomorphism of R such that us（u）ut = 0 for all u 2 L, where s; t are fixednon-negative integers. If either charR 〉 s ＋ t or charR = 0, then R satisfies s4, thestandard identity in four variables. We also examine the identity （σ（[x; y]）-[x; y]）n =0 for all x; y ∈ I, where I is a nonzero ideal of R and n is a fixed positive integer. Ifeither charR 〉 n or charR = 0, then R is commutative.

Left Annihilator of Power Values with Derivations on Lie Ideals in Prime Rings

Let R be a prime ring with characteristic di erent from two,d a derivation of R,L a noncentral Lie ideal of R,and a2R.In the present paper it is showed that if a(d(u^m)±u^m)^n for all u∈L,where m;n are xed positive integers,then a=0 unless R satis es s4,the standard polynomial identity in four variables.

Compositions of Generalized Derivations in Prime Rings

Let R be a prime ring with a non-central Lie ideal L. In the present paper we show that if the composition DG of two generalized derivations D and G is zero on L, then DG must be zero on R. And we get all the possibilities for the composition of a couple of generalized derivations to be zero on a non-central Lie ideal of a prime ring.

Annihilator on Co-commutators with Derivations on Lie Ideals in Prime Rings

Let R be a prime ring of characteristic different from 2, d and 9 two derivations of R at least one of which is nonzero, L a non-central Lie ideal of R, and a ∈ R. We prove that if a（d（u）u - ug（u）） = 0 for any u ∈ L, then either a = O, or R is an sa-ring, d（x） = [p, x], and g（x） = -d（x） for some p in the Martindale quotient ring of R.

(Right)Partial Generalized Automorphisms of Semiprime Rings

In this paper, we define the concept of (right) partial generalized automorphisms and discuss the extension problem, and also give a characterization of (right) partial generalized automorphisms of semiprime rings. Finally, we study the centralizing problem of right partial generalized automorphisms.

Centralizing Automorphisms of Lie Ideals in Prime Rings of Characteristic 2

Let R be a prime ring with an automorphism σ≠1, an identity map. Let L be a noncentral Lie ideal of R such that \xσ, x] ∈Z for all x ∈ L, where Z is the center of R. Then L is contained in the center of R, unless char(R) = 2 and dimcRC = 4.

Generalized Derivations Acting on Multilinear Polynomials in Prime Rings and Banach Algebras

Let R be a prime ring of characteristic different from 2 with Utumi quotientring U and extended centroid C,F and G,the two nonzero generalized derivationsof R,I an ideal of R and f(x1,...,xn)a multilinear polynomial over C which is notcentral valued on R.If F(G(f(x1,...,Xn))f(X1,...,Xn))=0 for all x1,...,Xn∈1,then one of the followings holds:(1)there exist a,b c Usuch that F(x)=ax and G(x)=bx for all x c R with ab=0;(2)there exista,b,p c U such that F(x)=ax+xb and G(x)=px for all x c R with F(p)=0and f(x1,...,xn)’is central valued on R.We also obtain some related results in caseswhere R is a semiprime ring and Banach algebra.

Strong Commutativity-preserving Generalized Derivations on Semiprime Rings

Let R be a ring with a subset S. A mapping of R into itself is called strong commutativitypreserving （scp） on S, if [f（x）, f（y）] = [x, y] for all x, y ∈ S. The main purpose of this paper is to describe the structure of the generalized derivations which are scp on some ideals and right ideals of a prime ring, respectively. The semiprime case is also considered.

Generalized Differential Identities of (Semi-)Prime Rings

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.

Centralizers of X-Generalized Skew Derivations on Multilinear Polynomials in Prime Rings

Let R be a prime ring of characteristic different from 2,Q_(r) be its right MartindalequotientringandC beitsextendedcentroid,G beanonzero X-generalized skew derivation of R,and S be the set of the evaluations of a multilinear polynomial f(x_(1),...,x_(n))over C with n non-commuting variables.Let u,v∈R be such that uG(x)x+G(x)xv=0 for all x∈S.Then one of the following statements holds:(a)v∈C and there exist a,b,c∈Q_(r) such that G(x)=ax+bxc for any x∈R with(u+v)a=(u+v)b=0;(b)f(x_(1),...,x_(n))2 is central-valued on R and there exists a∈Q r such that G(x)=ax for all x∈R with ua+av=0.

On Certain Functional Equation in Semiprime Rings

Let n be a fixed integer, let R be an （n ＋ 1）!-torsion free semiprime ring with the identity element and let F ： R → R be an additive mapping satisfying the relation F（xn＋2） ∑＋n＋i=1xi-1F（x）2xn＋1-i -∑xnF（x）xn＋1-i for all x 6 R. In this case, we prove that F is of the form 2F（x） = D（x） ＋ ax ＋ xa for all x ∈ R, where D ： R → R is a derivation and a 6 R is some fixed element.

Special Mappings with Central Values on Prime Rings

In this paper we investigate commutativity of prime rings with involution*of the second kind in which endomorphisms satisfy certain algebraic identities.Furthermore,we provide examples to show that the various restrictions imposed by the hypo theses of our theorems are not superfluous.

A note on generalized Lie derivations of prime rings

Let R be a prime ring of characteristic not 2, A be an additive subgroup of R, and F, T, D, K： A →R be additive maps such that F（[x, y]） = F（x）y - yg（x） - T（y）x ＋ xD（y） for all x, y ∈ A. Our aim is to deal with this functional identity when A is R itself or a noncentral Lie ideal of R. Eventually, we are able to describe the forms of the mappings F, T, D, and K in case A = R with deg（R） 〉 3 and also in the case A is a noncentral Lie ideal and deg（R） 〉 9. These enable us in return to characterize the forms of both generalized Lie derivations, D-Lie derivations and Lie centralizers of R under some mild assumptions. Finally, we give a generalization of Lie homomorphisms on Lie ideals.

On the Derivations of Semiprime Rings and Noncommutative Banach Algebras

Let A be a noncommutative Banach algebra.Suppose there exists a continuous linear Jordan derivation D:A→A such that [D(x),x]D(x)[D(x),x]∈ rad(A) for all x ∈ A.In this case, D(A)rad(A).

Some Results about Derivations of Prime Rings

Recently, basing on a suitable application of Milne-Bhatnagar＇s characterization theorem about matrix inversions, we have found that Warnaar＇s elliptic matrix inversion can be further extended to the following general inversion theorem.

Prime Rings with Generalized Derivations

The concept of derivations and generalized inner derivations has been generalized as an additive function δ： R → R satisfying δ （xy） = δ（x）y＋xd（y） for all x, y ∈ R, where d is a derivation on R. Such a function δ is called a generalized derivation. Suppose that U is a Lie ideal of R such that u^2 ∈ U for all u ∈ U. In this paper, we prove that U lahtain in Z（R） when one of the following holds： （1） δ（[u, v]） = u o v =（2） δ（[u,v]）=[u o v] = 0 （3） δ（u o v） = [u, v] （4） δ（u o v）＋δ[u, v] = 0 for all u, v ∈ U.