Commutators Having Idempotent Values with Automorphisms in Semi-Prime Rings (original) (raw)

An equation related to centralizers in semiprime rings

Glasnik Matematicki, 2003

The main result: Let R be a 2-torsion free semiprime ring with extended centroid C and let T : R → R be an additive mapping. Suppose that 3T (xyx) = T (x)yx + xT (y)x + xyT (x) holds for all x, y ∈ R. Then there exists an element λ ∈ C such that T (x) = λx for all x ∈ R. This research has been motivated by the work of Brešar [4] and Zalar [8]. Throughout, R will represent an associative ring with center Z (R). A ring R is n-torsion free, where n > 1 is an integer, in case nx = 0, x ∈ R implies x = 0. As usual the commutator xy − yx will be denoted by [x, y]. We shall use basis commutator identities [xy, z] = [x, z]y + x[y, z] and [x, yz] = [x, y] z + y [x, z]. Recall that R is prime if aRb = (0) implies a = 0 or b = 0, and is semiprime if aRa = (0) implies a = 0. An additive mapping D : R → R is called a derivation if D(xy) = D(x)y + xD(y) holds for all pairs x, y ∈ R and is called a Jordan derivation in case D(x 2) = D(x)x + xD(x) is fulfilled for all x ∈ R. A derivation D is inner in case there exists a ∈ R, such that D(x) = [a, x] holds for all x ∈ R. Every derivation is a Jordan derivation. The converse is in general not true. A classical result of Herstein [6] asserts that any Jordan derivation on a 2-torsion free prime ring is a derivation (see [2] for an alternative proof). Cusack [5] generalized Herstein's result on 2torsion free semiprime rings (see [3] for an alternative proof). We follow Zalar [8] and call an additive mapping T : R → R a left (right) centralizer in case T (xy) = T (x)y (T (xy) = xT (y)) holds for all x, y ∈ R. This concept appears naturally in C *-algebras. In ring theory it is more common to work with module homomorphisms. Ring theorists would write that T : R R → R R is

On Lie Ideals and Automorphisms in Prime Rings

Mathematical Notes, 2020

Let R be a prime ring of characteristic different from 2 with center Z and extended centroid C, and let L be a Lie ideal of R. Consider two nontrivial automorphisms α and β of R for which there exist integers m, n ≥ 1 such that α(u) n + β(u) m = 0 for all u ∈ L. It is shown that, under these assumptions, either L is central or R ⊆ M 2 (C) (where M 2 (C) is the ring of 2 × 2 matrices over C), L is commutative, and u 2 ∈ Z for all u ∈ L.

A Commutativity theorem for semiprime rings

Journal of the Australian Mathematical Society, 1980

It is shown that if R is a semiprime ring with 1 satisfying the property that, for each x, y e R, there exists a positive integer n depending on v and y such that (\_v)*-x*>'*is central for k = n,n+ 1,H + 2, then R is commutative, thus generalizing a result of Kaya.

Commutators with power central values on a Lie ideal

Pacific Journal of Mathematics, 2000

Let R be a prime ring of characteristic = 2 with a derivation d = 0, L a noncentral Lie ideal of R such that [d(u), u] n is central, for all u ∈ L. We prove that R must satisfy s 4 the standard identity in 4 variables. We also examine the case R is a 2-torsion free semiprime ring and [d([x, y]), [x, y]] n is central, for all x, y ∈ R.

On Centrally Prime and Centrally Semiprime Rings

AL-Rafidain Journal of Computer Sciences and Mathematics, 2008

In this paper, centrally prime and centrally semiprime rings are defined and the relations between these two rings and prime (resp. semiprime) rings are studied.Among the results of the paper some conditions are given under which prime (resp. semiprime) rings become centrally prime (resp.centrally semiprime) as in:1-A nonzero prime (resp. semiprime) ring which has no proper zero divisors is centrally prime (resp.centrally semiprime).Also we gave some other conditions which make prime (resp. semiprime) rings and centrally prime (resp.centrally semiprime) rings equivalent, as in :2-A ring which satisfies the-) (BZP for multiplicative systems is prime (resp. semiprime) if and only if it is centrally prime (resp.centrally semiprime).3-A ring with identity in which every nonzero element of its center is a unit is prime (resp. semiprime) if and only if it is centrally prime (resp.centrally semiprime).

On theta\theta theta-centralizers of semiprime rings (II)

St Petersburg Mathematical Journal, 2009

The following result is proved: Let R be a 2-torsion free semiprime ring, and let T : R → R be an additive mapping, related to a surjective homomorphism θ : R → R, such that 2T (x 2) = T (x)θ(x) + θ(x)T (x) for all x ∈ R. Then T is both a left and a right θ-centralizer. §1. Introduction This paper has been motivated by the work of Brešar [5], Vukman [10], and Zalar [11]. Throughout, R will represent an associative ring with center Z(R). Recall that R is prime if aRb = (0) implies a = 0 or b = 0, and R is semiprime if aRa = (0) implies a = 0. A ring R is 2-torsion free if 2x = 0, x ∈ R implies x = 0. As usual, the commutator xy − yx will be denoted by [x, y]. We shall use the commutator identities [x, yz] = [x, y]z + y[x, z] and [xy, z] = [x, z]y + x[y, z]. An additive mapping D : R → R is called a derivation if D(xy) = D(x)y + xD(y) for all pairs x, y ∈ R; D is called a Jordan derivation if D(x 2) = D(x)x + xD(x) for all x ∈ R. A derivation D is inner if there exists an element a ∈ R such that D(x) = [a, x] for all x ∈ R. An additive mapping D : R → R related to a map θ : R → R is called a (θ, θ)-derivation if D(xy) = D(x)θ(y) + θ(x)D(y) for all pairs x, y ∈ R; D is called a Jordan (θ, θ)-derivation if D(x 2) = D(x)θ(x) + θ(x)D(x) for all x ∈ R. A (θ, θ)-derivation D is inner if there exists a ∈ R such that D(x) = [a, θ(x)] for all x ∈ R. It is clear that if θ is the identity map on R, then every (θ, θ)-derivation is an ordinary derivation. Every derivation is a Jordan derivation; the converse is in general not true. A classical result of Herstein [7] asserts that any Jordan derivation on a 2-torsion free prime ring is a derivation. A brief proof of Herstein's result can be found in [2]. Cusack [6] generalized Herstein's result to 2-torsion free semiprime rings (see also [4] for an alternative proof). Zalar [11] gave the following definition: An additive mapping T : R → R is called a left (right) centralizer if T (xy) = T (x)y (T (xy) = xT (y)) for all x, y ∈ R. If R is a ring with involution * , then every additive mapping E : R → R that satisfies E(x 2) = E(x)x * + xE(x) for all x ∈ R is called a Jordan *-derivation. These mappings are closely related to the question of the representability of quadratic forms by bilinear forms. Some algebraic properties of Jordan *-derivations were considered in [3], where further references can be found. For quadratic forms, see [9]. If the product in R is given by x • y = xy + yx, then a Jordan derivation is an additive mapping D satisfying D(x • y) = D(x) • y + x • D(y) for all x, y ∈ R; a Jordan homomorphism is an additive mapping A satisfying A(x • y) = A(x) • A(y) for all x, y ∈ R. Zalar [11] defined the Jordan centralizer to be an additive mapping T such that T (x • y) = T (x) • y = x • T (y). Since the product • is commutative, there is no difference between the left and the right Jordan centralizers.

On a Class of Semicommutative Rings

2017

Let R be a ring with identity and an ideal I. In this paper, we introduce a class of rings generalizing semicommutative rings which is called I-semicommutative. The ring R is called I-semicommutative whenever ab = 0 implies aRb ⊆ I for any a, b ∈ R. We investigate general properties of I-semicommutative rings and show that several results of semicommutative rings and J-semicommutative rings can be extended to I-semicommutative rings for this general settings.

On Unit-Central Rings

2015

Dedicated to S. K. Jain in honor of his 70th birthday. Abstract. We establish commutativity theorems for certain classes of rings in which every invertible element is central, or, more generally, in which all invertible elements commute with one another. We prove that if R is a semiex-change ring (i.e. its factor ring modulo its Jacobson radical is an exchange ring) with all invertible elements central, then R is commutative. We also prove that if R is a semiexchange ring in which all invertible elements com-mute with one another, and R has no factor ring with two elements, then R is commutative. We offer some examples of noncommutative rings in which all invertible elements commute with one another, or are central. We close with a list of problems for further research.