Nadeem ur Rehman

On Commutativity of Rings With Derivations

Let R be a ring and d : R → R a derivation of R. In the present paper we investigate commutativity of R satisfying any one of the properties (i)d([x,y]) = [x,y], (ii)d(x o y) = xoy, (iii)d(x) o d(y) = 0, or (iv)d(x) o d(y) = x o y, for all x, y in some apropriate subset of R.

On Generalized (α,β)-Derivations in Prime Rings

Let R be a ring and α,β be endomorphisms of R. An additive mapping F: R → R is called a generalized (α,β)-derivation on R if there exists an (α,β)-derivation d: R → R such that F(xy)=F(x) α(y) + β(x)d(y) holds for all x, y ∈ R. In the present paper, we discuss the commutativity of a prime ring R admitting a generalized (α,β)-derivation F satisfying any one of the properties: (i) [F(x),x]α,β=0, (ii) F([x,y])=0, (iii) F(x ◦ y)=0, (iv) F([x,y])=[x,y]α,β, (v) F(x ◦ y)=(x ◦ y)α,β, (vi) F(xy)- α(xy) ∈ Z(R), (vii) F(x)F(y)- α(xy) ∈ Z(R) for all x, y in an appropriate subset of R.

IDENTITIES WITH ADDITIVE MAPPINGS IN SEMIPRIME RINGS

Abstract. The aim of this paper is to prove the next result. Let n > 1be an integer and let R be a n!-torsion free semiprime ring. Suppose thatf : R →R is an additive mapping satisfying the relation [f(x),x n ] = 0for all x ∈R. Then f is commuting on R. 1. Introduction and the maintheoremThroughout, R will represent an associative ring with a center Z(R). Letn > 1 be an integer. A ring R is n-torsion free if nx = 0, x ∈ R, impliesx = 0. The Lie product (or a commutator) of elements x,y ∈ R will be denotedby [x,y] (i.e., [x,y] = xy − yx). Recall that a ring R is prime if aRb = {0},a,b ∈ R, implies that either a = 0 or b = 0. Furthermore, a ring R is calledsemiprime if aRa = {0}, a ∈ R, implies a = 0. We will denote by C and Q theextended centroid and the maximal right ring of quotients of a semiprime ringR, respectively. For the explanation of the extended centroid as well as themaximal right ring of quotients of a semiprime ring we refer the reader to [4].As usual, the socle of a ring R will be denoted by soc(R).An additive mapping D : R → R is called a derivation on R if D(xy) =D(x)y + xD(y) holds for all pairs x,y ∈ R. An additive mapping f : R →R is called centralizing on R if [f(x),x] ∈ Z(R) holds for all x ∈ R. In aspecial case, when [f(x),x] = 0 for all x ∈ R, the mapping f is said to becommuting on R. A classical result of Posner [21] (Posner’s second theorem)states that the existence of a nonzero centralizing derivation on a prime ringforces the ring to be commutative. Posner’s second theorem in general cannotbe proved for semiprime rings as shows the following example. Let R

A Note on Skew-commuting Automorphisms in Prime Rings

Let R be a prime ring with center Z, I a nonzero ideal of R, and a non- trivial automorphism of R such that {(x ◦ y) − (x ◦ y)} n ∈ Z for all x;y ∈ I. If either char(R) > n or char (R) = 0, then R satises s4, the standard identity in 4 variables.

Commutativity and Skew-commutativity Conditions with Generalized Derivations

Let R be a prime ring of characteristic different from 2 with extended centroid C. Let F be a generalized derivation of R, L a non-central Lie ideal of R, f(x1, …, xn) a polynomial over C and f(R)={f(r1, …, rn): ri ∈ R}. We study the following cases: (1) [F(u), F(v)]k=0 for all u, v ∈ L, where k ≥ 1 is a fixed integer; (2) [F(u), F(v)] = 0 for all u, v ∈ f(R); (3) F(u) ◦ F(v)=0 for all u, v ∈ f(R); (4) F(u) ◦ F(v)=u ◦ v for all u, v ∈ f(R). We obtain a description of the structure of R and information on the form of F.

On skew-commuting mappings in semiprime rings

Abstract In this paper we prove the following result. Let R be a n!-torsion free semiprime ring and let f :R → R be an additive mapping satisfying the relation f(x)xn + xnf(x) = 0 for all x ∈ R. In this case f = 0.

LIE IDEALS, MORITA CONTEXT AND GENERALIZED ( α , β )-DERIVATIONS

Abstract A classical problem in ring theory is to study conditions under which a ring is forced to become commutative. Stimulated from Jacobsons famous result, several techniques are developed to achieve this goal. In the present note, we use a pair of rings, which are the ingredients of a Morita context, and obtain that if one of the ring is prime with the generalized (α, β)-derivations that satisfy certain conditions on the trace ideal of the ring, which by default is a Lie ideal, and the other ring is reduced, then the trace ideal of the reduced ring is contained in the center of the ring. As an outcome, in case of a semi-projective Morita context, the reduced ring becomes commutative.

A note on multiplicative (generalized)-skew derivation on semiprime rings

ABSTRACT In this article, we study multiplicative (generalized)-skew derivation G and multiplicative left centralizer H satisfying certain conditions in semiprime rings. Moreover, some examples are given to demonstrate that the semiprimeness imposed on the hypotheses of various theorems is essential.

Identity related to additive mappings on standard operator algebras

Abstract Let X be a real or complex Banach space, let be the algebra of all bounded linear operators of X and let be a standard operator algebra. Suppose there exists a linear mapping satisfying the relation T(An) = T(A)An−1 − AT(An−2)A − An−1T(A) for all , where n > 2 is some fixed integer. Then T is of the form: (i)T(A) = 0 for all and (ii) T(A) = BA, for all and some .

Centralizing mappings, Morita context and generalized (α, β)-derivations

Abstract In the present note, we use a pair of rings, which are the ingredients of a Morita context, and obtain that if one of the ring is prime with the generalized (α, β)-derivations that satisfy certain conditions on the trace ideal of the ring, which by default is a Lie ideal, and the other ring is reduced, then the trace ideal of the reduced ring is contained in the center of the ring. As an outcome, in case of a semi-projective Morita context, the reduced ring becomes commutative.