Derivations on a Lie Ideal

1988 ◽  
Vol 31 (3) ◽  
pp. 280-286 ◽  
Author(s):  
Silvana Mauceri ◽  
Paola Misso
Keyword(s):  
Prime Ring ◽  
Division Ring ◽  
Lie Ideal ◽  
Inner Derivation

AbstractIn this paper we prove the following result: let R be a prime ring with no non-zero nil left ideals whose characteristic is different from 2 and let U be a non central Lie ideal of R.If d ≠ 0 is a derivation of R such that d(u) is invertible or nilpotent for all u ∈ U, then either R is a division ring or R is the 2 X 2 matrices over a division ring. Moreover in the last case if the division ring is non commutative, then d is an inner derivation of R.

Derivations with Invertible Values on a Lie Ideal

1988 ◽  
Vol 31 (1) ◽  
pp. 103-110 ◽  
Author(s):  
Jeffrey Bergen ◽  
L. Carini
Keyword(s):  
Division Ring ◽  
Unit Element ◽  
Lie Ideal ◽  
Division Rings

AbstractLet R be a ring which possesses a unit element, a Lie ideal U ⊄ Z, and a derivation d such that d(U) ≠ 0 and d(u) is 0 or invertible, for all u ∈ U. We prove that R must be either a division ring D or D2, the 2 X 2 matrices over a division ring unless d is not inner, R is not semiprime, and either 2R or 3R is 0. We also examine for which division rings D, D2 can possess such a derivation and study when this derivation must be inner.

Derivations with annihilator conditions on Lie ideals in prime rings

2019 ◽  
Vol 19 (02) ◽  
pp. 2050025 ◽  
Author(s):  
Shuliang Huang
Keyword(s):  
Prime Ring ◽  
Polynomial Identity ◽  
Lie Ideal ◽  
Prime Rings ◽  
Positive Integers

Let [Formula: see text] be a prime ring with characteristic different from two, [Formula: see text] a derivation of [Formula: see text], [Formula: see text] a noncentral Lie ideal of [Formula: see text], and [Formula: see text]. In the present paper, it is shown that if one of the following conditions holds: (i) [Formula: see text], (ii) [Formula: see text], (iii) [Formula: see text] and (iv) [Formula: see text] for all [Formula: see text], where [Formula: see text] are fixed positive integers, then [Formula: see text] unless [Formula: see text] satisfies [Formula: see text], the standard polynomial identity in four variables.

scholarly journals Derivations on Lie Ideals of σ-Prime Γ-Rings

2015 ◽  
Vol 39 (2) ◽  
pp. 249-255
Author(s):  
Md Mizanor Rahman ◽  
Akhil Chandra Paul
Keyword(s):  
Prime Ring ◽  
Additive Mapping ◽  
Jordan Derivation ◽  
Lie Ideal ◽  
Square Closed Lie Ideal ◽  
Academy Of Sciences ◽  
Gamma Rings ◽  
Torsion Free

The authors extend and generalize some results of previous workers to ?-prime ?-ring. For a ?-square closed Lie ideal U of a 2-torsion free ?-prime ?-ring M, let d: M ?M be an additive mapping satisfying d(u?u)=d(u)? u + u?d(u) for all u ? U and ? ? ?. The present authors proved that d(u?v) = d(u)?v + u?d(v) for all u, v ? U and ?? ?, and consequently, every Jordan derivation of a 2-torsion free ?-prime ?-ring M is a derivation of M.Journal of Bangladesh Academy of Sciences, Vol. 39, No. 2, 249-255, 2015

scholarly journals Jordan Derivations on Lie Ideals of ?-Prime Rings

2017 ◽  
Vol 36 ◽  
pp. 1-5
Author(s):  
Akhil Chandra Paul ◽  
Md Mizanor Rahman
Keyword(s):  
Prime Ring ◽  
Additive Mapping ◽  
Lie Ideal ◽  
Prime Rings ◽  
R And D ◽  
Square Closed Lie Ideal ◽  
Torsion Free

In this paper we prove that, if U is a s-square closed Lie ideal of a 2-torsion free s-prime ring R and  d: R(R is an additive mapping satisfying d(u2)=d(u)u+ud(u) for all u?U then d(uv)=d(u)v+ud(v) holds for all  u,v?UGANIT J. Bangladesh Math. Soc.Vol. 36 (2016) 1-5

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.

scholarly journals Generalised Inner Derivations in Semi Prime Rings

2020 ◽  
Vol 5 (7) ◽  
pp. 372-374
Author(s):  
Dr. K. L Kaushik
Keyword(s):  
Prime Ring ◽  
Inner Derivation ◽  
Prime Rings ◽  
Fixed Element

Let A be any ring and f(xy) = f(x)y+xha(y), where f be any generalised inner derivation(G.I.D ) a be the fixed element of A. In this paper, it is shown that (i) ha must necessarily be a derivation for semi prime ring A. (ii) ∃ no generalized inner derivations f : A → A such that f(x ◦ y) = x ◦ y or f(x ◦ y) + x ◦ y = 0 ∀ x,y ∈ A, We have proved Havala [2] def. p.1147, Herstein [3] Lemma 3.1 p. 1106 as corollaries, along with other results.

scholarly journals Posner’s second theorem and some related annihilating conditions on lie ideals

Filomat ◽  
2018 ◽  
Vol 32 (4) ◽  
pp. 1285-1301 ◽  
Author(s):  
Emine Albaş ◽  
Nurcan Argaç ◽  
Filippis de
Keyword(s):  
Prime Ring ◽  
Quotient Ring ◽  
Lie Ideal ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Utumi Quotient Ring

Let R be a non-commutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, L a non-central Lie ideal of R, F and G two non-zero generalized derivations of R. If [F(u),u]G(u) = 0 for all u ? L, then one of the following holds: (a) there exists ? ? C such that F(x) = ?x, for all x ? R; (b) R ? M2(F), the ring of 2 x 2 matrices over a field F, and there exist a ? U and ? ? C such that F(x) = ax + xa + ?x, for all x ? R.

On Posner’s theorem with b-generalized skew derivations on Lie ideals

2021 ◽  
Author(s):  
Luisa Carini ◽  
Giovanni Scudo
Keyword(s):  
Prime Ring ◽  
Quotient Ring ◽  
Lie Ideal ◽  
Extended Centroid ◽  
Skew Derivation ◽  
Generalized Skew Derivation ◽  
Martindale Quotient Ring

Let [Formula: see text] be a non-commutative prime ring of characteristic different from [Formula: see text] and [Formula: see text], [Formula: see text] its right Martindale quotient ring and [Formula: see text] its extended centroid. Suppose that [Formula: see text] is a non-central Lie ideal of [Formula: see text], [Formula: see text] a nonzero [Formula: see text]-generalized skew derivation of [Formula: see text]. If [Formula: see text] for all [Formula: see text], then one of the following holds: (a) there exists [Formula: see text] such that [Formula: see text], for all [Formula: see text]; (b) [Formula: see text], the ring of [Formula: see text] matrices over [Formula: see text], and there exist [Formula: see text] and [Formula: see text] such that [Formula: see text], for all [Formula: see text].

scholarly journals Some notes on Lie ideals in division rings

2018 ◽  
Vol 17 (03) ◽  
pp. 1850049
Author(s):  
M. Aaghabali ◽  
M. Ariannejad ◽  
A. Madadi
Keyword(s):  
Division Ring ◽  
Commutator Subgroup ◽  
Product Formula ◽  
Main Concern ◽  
Normal Subgroups ◽  
Lie Ideal ◽  
Additive Subgroup ◽  
Finitely Generated ◽  
Finite Dimensional ◽  
Lie Product Formula

A Lie ideal of a division ring [Formula: see text] is an additive subgroup [Formula: see text] of [Formula: see text] such that the Lie product [Formula: see text] of any two elements [Formula: see text] is in [Formula: see text] or [Formula: see text]. The main concern of this paper is to present some properties of Lie ideals of [Formula: see text] which may be interpreted as being dual to known properties of normal subgroups of [Formula: see text]. In particular, we prove that if [Formula: see text] is a finite-dimensional division algebra with center [Formula: see text] and [Formula: see text], then any finitely generated [Formula: see text]-module Lie ideal of [Formula: see text] is central. We also show that the additive commutator subgroup [Formula: see text] of [Formula: see text] is not a finitely generated [Formula: see text]-module. Some other results about maximal additive subgroups of [Formula: see text] and [Formula: see text] are also presented.

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].

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