Generalized (Jordan) left derivations on rings associated with an element of rings
type: Item
instance of: scholarly article
title: Generalized (Jordan) left derivations on rings associated with an element of rings
zbMATH Open document ID: 1387.16040
DOI: 10.3103/S1068362317040021
author: L. K. Ardekani
author: Bijan Davvaz
published in: Journal of Contemporary Mathematical Analysis. Armenian Academy of Sciences
publication date: 2017-11-06T00:00:00Z
review text: In this paper, the authors introduce a new notion of generalized (Jordan) left derivation on rings as follows; let \(R\) be a ring, an additive mapping \(F: R\to R\) is called a generalized (resp., Jordan) left derivation if there exists an element \(w\in R\) such that \(F(xy) = xF(y) + yF(x) + yxw\) (resp., \(F(x^2) = 2xF(x) + x^2w)\) for all \(x,y\in R\). In addition, some related properties and results on generalized (Jordan) left derivation of square closed Lie ideals are obtained. In fact, the authors focus on Lemma 2.1, which contains 6 parts, but they leave it without proof. I believe the paper would be better if they would have provide at least two of these parts. However, the authors mention good background information. To become more closely to definition of generalized (Jordan) left derivations on rings the authors supply some examples to illustrate it. In the proof of Theorem 2.1, I believe something is not completely correct, since the authors rely on Lemma 2.1, stating that \((F,w):R\to R\) is a generalized Jordan left derivation on a square closed Lie ideal of \(R\), while Theorem 2.1 mentions that \((F,w):R\to R\) is a generalized Jordan left derivation on a 2-torsion free semiprime ring \(R\). The authors apply this result of the proof for another theorems.
reviewed by: Mehsin Atteya
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16Y99
Mathematics Subject Classification ID: 20N20
zbMATH DE Number: 6803065
zbMATH Keywords: ring
zbMATH Keywords: generalized left derivation
zbMATH Keywords: generalized Jordan left derivation
zbMATH Keywords: square closed Lie idea
MaRDI profile type: Publication
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full work available at URL: s1068362317040021
OpenAlex ID: W2757889714
description: scientific article; zbMATH DE number 6803065