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.

On Automorphisms and Commutativity in Semiprime Rings

2013 ◽  
Vol 56 (3) ◽  
pp. 584-592 ◽  
Author(s):  
Pao-Kuei Liau ◽  
Cheng-Kai Liu
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Identity Automorphism ◽  
Semiprime Rings ◽  
Positive Integers

Abstract. Let R be a semiprime ring with center Z(R). For x, y ∊ R, we denote by [x, y] = xy – yx the commutator of x and y. If σ is a non-identity automorphism of R such thatfor all x ∊ R, where n0, n1, n2, … nk are fixed positive integers, then there exists a map μ: R → Z(R) such that σ(x) = x + μ(x) for all x ∊ R. In particular, when R is a prime ring, R is commutative.

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.

Semiprime Rings with Hypercentral Derivations

1995 ◽  
Vol 38 (4) ◽  
pp. 445-449 ◽  
Author(s):  
Tsiu-Kwen Lee
Keyword(s):  
Left Ideal ◽  
Semiprime Ring ◽  
Quotient Ring ◽  
Central Idempotent ◽  
Utumi Quotient Ring ◽  
Semiprime Rings ◽  
Positive Integers

AbstractLetRbe a semiprime ring with a derivationd, λ a left ideal ofRandk, ntwo positive integers. Suppose that[d(xn),xn]k= 0 for allx∊ λ. Then [λ,R]d(R)= 0. That is, there exists a central idempotente∊U, the left Utumi quotient ring ofR, such thatdvanishes identically oneUand λ(l —e) is central in (1 —e)U

Generalized derivations preserving Engel condition in prime and semiprime rings

2020 ◽  
pp. 2150041
Author(s):  
Rita Prestigiacomo
Keyword(s):  
Prime Ring ◽  
Algebraic Closure ◽  
Quotient Ring ◽  
Lie Ideal ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Martindale Quotient Ring ◽  
Engel Condition

Let [Formula: see text] be a prime ring with [Formula: see text], [Formula: see text] a non-central Lie ideal of [Formula: see text], [Formula: see text] its Martindale quotient ring and [Formula: see text] its extended centroid. Let [Formula: see text] and [Formula: see text] be nonzero generalized derivations on [Formula: see text] such that [Formula: see text] Then there exists [Formula: see text] such that [Formula: see text] and [Formula: see text], for any [Formula: see text], unless [Formula: see text], where [Formula: see text] is the algebraic closure of [Formula: see text].

A note on an Engel condition with derivations in rings

2017 ◽  
Vol 26 (1) ◽  
pp. 19-27
Author(s):  
Mohd Arif Raza ◽  
◽  
Nadeem Rehman ur ◽  
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Extended Centroid ◽  
Engel Condition ◽  
Positive Integers

Let $R$ be a prime ring with center $Z(R)$, $C$ the extended centroid of $R$, $d$ a derivation of $R$ and $n,k$ be two fixed positive integers. In the present paper we investigate the behavior of a prime ring $R$ satisfying any one of the properties (i)~$d([x,y]_k)^n=[x,y]_k$ (ii) if $char(R)\neq 2$, $d([x,y]_k)-[x,y]_k\in Z(R)$ for all $x,y$ in some appropriate subset of $R$. Moreover, we also examine the case when $R$ is a semiprime ring

Antiautomorphisms with quasi-generalized Engel condition

2018 ◽  
Vol 17 (08) ◽  
pp. 1850145 ◽  
Author(s):  
M. Chacron
Keyword(s):  
Finite Order ◽  
Prime Ring ◽  
Division Ring ◽  
Idempotent Element ◽  
General Question ◽  
Engel Condition ◽  
Positive Integers

Let [Formula: see text] be a ring with 1. Given elements [Formula: see text], [Formula: see text] of [Formula: see text] and the integer [Formula: see text] define [Formula: see text] and [Formula: see text]. We say that a given antiautomorphism [Formula: see text] of [Formula: see text] is commuting if [Formula: see text], all [Formula: see text]. More generally, assume that [Formula: see text] satisfies the condition [Formula: see text] where [Formula: see text], [Formula: see text] are corresponding positive integers depending on [Formula: see text], and [Formula: see text] ranges over [Formula: see text]. To what extent can one say that [Formula: see text] is commuting? In this paper, we answer the question in the affirmative if R is a prime ring containing some idempotent element [Formula: see text]. In the diametrically opposed case in which [Formula: see text] is a division ring the answer is again yes provided [Formula: see text] is algebraic over its center and [Formula: see text] is of finite order. These two major complementary results will be put to work to provide an answer to the general question.

An Engel condition with b-generalized derivations for Lie ideals

2018 ◽  
Vol 17 (03) ◽  
pp. 1850046 ◽  
Author(s):  
Pao-Kuei Liau ◽  
Cheng-Kai Liu
Keyword(s):  
Prime Ring ◽  
Analogous Result ◽  
Matrix Ring ◽  
Generalized Derivation ◽  
Lie Ideal ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Engel Condition ◽  
Positive Integers

Let [Formula: see text] be a prime ring with the extended centroid [Formula: see text], [Formula: see text] a noncommutative Lie ideal of [Formula: see text] and [Formula: see text] a nonzero [Formula: see text]-generalized derivation of [Formula: see text]. For [Formula: see text], let [Formula: see text]. We prove that if [Formula: see text] for all [Formula: see text], where [Formula: see text] are fixed positive integers, then there exists [Formula: see text] such that [Formula: see text] for all [Formula: see text] except when [Formula: see text], the [Formula: see text] matrix ring over a field [Formula: see text]. The analogous result for generalized skew derivations is also described. Our theorems naturally generalize the cases of derivations and skew derivations obtained by Lanski in [C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), 75–80, Skew derivations and Engel conditions, Comm. Algebra 42 (2014), 139–152.]

Lectures on von Neumann Algebras

2019 ◽  
Author(s):  
Serban-Valentin Stratila ◽  
Laszlo Zsido
Keyword(s):  
Von Neumann Algebras ◽  
Von Neumann
Sign in / Sign up

Export Citation Format

Share Document