scholarly journals Identities with derivations and automorphisms on semiprime rings

2005 ◽  
Vol 2005 (7) ◽  
pp. 1031-1038 ◽  
Author(s):  
Joso Vukman
Keyword(s):  
Prime Ring ◽  
Classical Result ◽  
Semiprime Rings

The purpose of this paper is to investigate identities with derivations and automorphisms on semiprime rings. A classical result of Posner states that the existence of a nonzero centralizing derivation on a prime ring forces the ring to be commutative. Mayne proved that in case there exists a nontrivial centralizing automorphism on a prime ring, then the ring is commutative. In this paper, some results related to Posner's theorem as well as to Mayne's theorem are proved.

scholarly journals A result related to derivations on unital semiprime rings

2021 ◽  
Vol 56 (1) ◽  
pp. 95-106
Author(s):  
Irena Kosi-Ulbl ◽  
◽  
Nejc Širovnik ◽  
Joso Vukman ◽  
◽  
...  
Keyword(s):  
Prime Ring ◽  
Classical Result ◽  
Additive Mapping ◽  
Jordan Derivation ◽  
Fixed Integer ◽  
Unital Ring ◽  
Semiprime Rings ◽  
History Of ◽  
Torsion Free

The purpose of this paper is to prove the following result. Let n≥3 be some fixed integer and let R be a (n+1)!2n-2-torsion free semiprime unital ring. Suppose there exists an additive mapping D: R→ R satisfying the relation for all x ∈ R. In this case D is a derivation. The history of this result goes back to a classical result of Herstein, which states that any Jordan derivation on a 2-torsion free prime ring is a derivation.

A Note on Differential Identities in Prime and Semiprime Rings

2020 ◽  
pp. 77-83
Author(s):  
Mohammad Shadab Khan ◽  
Mohd Arif Raza ◽  
Nadeemur Rehman
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Nonzero Ideal ◽  
Differential Identities ◽  
Standard Identity ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Positive Integers

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d ( r ○ s)(r ○ s) + ( r ○ s) d ( r ○ s)n - d ( r ○ s))m for all r, s ϵ I, then R is commutative. (ii) If (d ( r ○ s)( r ○ s) + ( r ○ s) d ( r ○ s)n - d (r ○ s))m ϵ Z(R) for all r, s ϵ I, then R satisfies s4, the standard identity in four variables. Moreover, we also examine the case when R is a semiprime ring.

scholarly journals Commutativity results for semiprime rings with derivations

1998 ◽  
Vol 21 (3) ◽  
pp. 471-474 ◽  
Author(s):  
Mohammad Nagy Daif
Keyword(s):  
Prime Ring ◽  
Semiprime Rings

We extend a result of Herstein concerning a derivationdon a prime ringRsatisfying[d(x),d(y)]=0for allx,y∈R, to the case of semiprime rings. An extension of this result is proved for a two-sided ideal but is shown to be not true for a one-sided ideal. Some of our recent results dealing withU*- andU**- derivations on a prime ring are extended to semiprime rings. Finally, we obtain a result on semiprime rings for whichd(xy)=d(yx)for allx,yin some idealU.

On commuting additive mappings on semiprime rings

2020 ◽  
pp. 2150079
Author(s):  
Siriporn Lapuangkham ◽  
Utsanee Leerawat
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Semiprime Rings ◽  
Additive Mappings

The main purpose of this paper is to describe the structure of a pair of additive mappings that are commuting on a semiprime ring. Furthermore, we prove that the existence of different commuting epimorphisms on a prime ring forces the ring to be commutative. Finally, we characterize additive mappings, which act as homomorphisms or antihomomorphisms on a semiprime ring.

AN ENGEL CONDITION WITH AUTOMORPHISMS FOR LEFT IDEALS

2013 ◽  
Vol 13 (02) ◽  
pp. 1350092 ◽  
Author(s):  
CHENG-KAI LIU
Keyword(s):  
Left Ideal ◽  
Prime Ring ◽  
Von Neumann Algebras ◽  
Von Neumann ◽  
Identity Automorphism ◽  
Semiprime Rings ◽  
Engel Condition ◽  
Positive Integers

Let R be a prime ring and L a nonzero left ideal of R. For x, y ∈ R, we denote [x, y] = xy-yx the commutator of x and y. In this paper, we prove that if R admits a non-identity automorphism σ such that [[…[[σ(xn0), xn1], xn2], …], xnk] = 0 for all x ∈ L, where n0, n1, n2, …, nk are fixed positive integers, then R is commutative. The analogous results for semiprime rings and von Neumann algebras are also obtained.

scholarly journals Commutatively of Prime and Semiprime ?-Rings with Symmetric BI-Derivations

2016 ◽  
Vol 34 ◽  
pp. 27-33
Author(s):  
Kalyan Kumar Dey ◽  
Akhil Chandra Paul
Keyword(s):  
Prime Ring ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Torsion Free

Let M be a ?-ring and let D: M x M ->M be a symmetric bi-derivation with the trace d: M -> M denoted by d(x) = D(x, x) for all x?M. The objective of this paper is to prove some results concerning symmetric bi-derivation on prime and semiprime ?-rings. If M is a 2-torsion free prime ?-ring and D ? 0 be a symmetric bi-derivation with the trace d having the property d(x)?x - x?d(x) = 0 for all x?M and ???, then M is commutative. We also prove another result in ?-rings setting analogous to that of Posner for prime rings.GANIT J. Bangladesh Math. Soc.Vol. 34 (2014) 27-33

scholarly journals Symmetric generalized biderivations on prime rings

2021 ◽  
Vol 39 (4) ◽  
pp. 65-72
Author(s):  
Faiza Shujat
Keyword(s):  
Prime Ring ◽  
Prime Rings ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings

The purpose of the present paper is to prove some results concerning symmetric generalized biderivations on prime and semiprime rings which partially extend some results of Vukman \cite {V}. Infact we prove that: let $R$ be a prime ring of characteristic not two and $I$ be a nonzro ideal of $R$. If $\Delta$ is a symmetric generalized biderivation on $R$ with associated biderivation $D$ such that $[\Delta(x,x), \Delta(y,y)]=0$ for all $x,y \in I$, then one of the following conditions hold\\ \begin{enumerate} \item $R$ is commutative. \item $\Delta$ acts as a left bimultiplier on $R$. \end{enumerate}

scholarly journals A pair of generalized derivations in prime, semiprime rings and in Banach algebras

2021 ◽  
Vol 39 (4) ◽  
pp. 131-141
Author(s):  
Basudeb Dhara ◽  
Venus Rahmani ◽  
Shervin Sahebi
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Banach Algebras ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Fixed Integer ◽  
Semiprime Rings ◽  
Commutative Banach Algebras

Let R be a prime ring with extended centroid C, I a non-zero ideal of R and n ≥ 1 a fixed integer. If R admits the generalized derivations H and G such that (H(xy)+G(yx))n= (xy ±yx) for all x,y ∈ I, then one ofthe following holds:(1) R is commutative;(2) n = 1 and H(x) = x and G(x) = ±x for all x ∈ R.Moreover, we examine the case where R is a semiprime ring. Finally, we apply the above result to non-commutative Banach algebras.

scholarly journals On n-generalized commutators and Lie ideals of rings

2021 ◽  
pp. 2250221
Author(s):  
Peter V. Danchev ◽  
Tsiu-Kwen Lee
Keyword(s):  
Positive Integer ◽  
Prime Ring ◽  
Classical Result ◽  
Associative Ring ◽  
Nonzero Ideal ◽  
Lie Ideal ◽  
Additive Subgroup ◽  
Simple Ring ◽  
Noncommutative Polynomials ◽  
Generalized Commutator

Let [Formula: see text] be an associative ring. Given a positive integer [Formula: see text], for [Formula: see text] we define [Formula: see text], the [Formula: see text]-generalized commutator of [Formula: see text]. By an [Formula: see text]-generalized Lie ideal of [Formula: see text] (at the [Formula: see text]th position with [Formula: see text]) we mean an additive subgroup [Formula: see text] of [Formula: see text] satisfying [Formula: see text] for all [Formula: see text] and all [Formula: see text], where [Formula: see text]. In the paper, we study [Formula: see text]-generalized commutators of rings and prove that if [Formula: see text] is a noncommutative prime ring and [Formula: see text], then every nonzero [Formula: see text]-generalized Lie ideal of [Formula: see text] contains a nonzero ideal. Therefore, if [Formula: see text] is a noncommutative simple ring, then [Formula: see text]. This extends a classical result due to Herstein [Generalized commutators in rings, Portugal. Math. 13 (1954) 137–139]. Some generalizations and related questions on [Formula: see text]-generalized commutators and their relationship with noncommutative polynomials are also discussed.

Permuting Jordan Left Tri – Derivations On Prime and semiprime rings

2018 ◽  
Vol 10 (3) ◽  
Author(s):  
Mazen O. Karim
Keyword(s):  
Prime Ring ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Torsion Free

             Let  be a 2 and 3 – torsion free prime ring then  if  admits a non-zero Jordan  left tri- derivation   ,   then  is commutative ,also we give some properties of permuting left tri - derivations.

Sign in / Sign up

Export Citation Format

Share Document