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Generalized Jordan Semiderivations in Prime Rings

Published online by Cambridge University Press:  20 November 2018

Vincenzo De Filippis ,
Abdellah Mamouni  and
Lahcen Oukhtite
Vincenzo De Filippis
Affiliation:
Department of Mathematics and Computer Science, University of Messina, 98166, Messina, Italy. e-mail: [email protected]
Abdellah Mamouni
Affiliation:
Université Moulay Ismail, Faculté des Sciences et Techniques Département de Mathématiques, BP. 509- Boutalamine 52000 Errachidia, Maroc. e-mail: [email protected], e-mail: [email protected] [email protected]
Lahcen Oukhtite
Affiliation:
Université Moulay Ismail, Faculté des Sciences et Techniques Département de Mathématiques, BP. 509- Boutalamine 52000 Errachidia, Maroc. e-mail: [email protected], e-mail: [email protected] [email protected]
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Abstract

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Let $R$ be a ring and let $g$ be an endomorphism of $R$. The additive mapping $d:\,R\,\to \,R$ is called a Jordan semiderivation of $R$, associated with $g$, if

$$d\left( {{x}^{2}} \right)=d\left( x \right)x+g\left( x \right)d\left( x \right)=d\left( x \right)g\left( x \right)+xd\left( x \right)\,\text{and}\,d\left( g\left( x \right) \right)=g\left( d\left( x \right) \right)$$

for all $x\,\in \,R$. The additive mapping $F:\,R\,\to \,R$ is called a generalized Jordan semiderivation of $R$, related to the Jordan semiderivation $d$ and endomorphism $g$, if

$$F\left( {{x}^{2}} \right)=F\left( x \right)x+g\left( x \right)d\left( x \right)=F\left( x \right)g\left( x \right)+xd\left( x \right)\,\,and\,F\left( g\left( x \right) \right)=g\left( F\left( x \right) \right)$$

for all $x\,\in \,R$. In this paper we prove that if $R$ is a prime ring of characteristic different from 2, $g$ an endomorphism of $R,\,d$ a Jordan semiderivation associated with $g,\,F$ a generalized Jordan semiderivation associated with $d$ and $g$, then $F$ is a generalized semiderivation of $R$ and $d$ is a semiderivation of $R$. Moreover, if $R$ is commutative, then $F\,=\,d$.

Keywords

16W25semiderivationgeneralized semiderivationJordan semiderivationprime ring
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 2 , 01 June 2015 , pp. 263 - 270
Copyright
Copyright © Canadian Mathematical Society 2015

References

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