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# Generalized Jordan Semiderivations in Prime Rings

Published:2015-02-06
Printed: Jun 2015
• Vincenzo De Filippis,
Department of Mathematics and Computer Science, University of Messina, 98166, Messina, Italy
• Abdellah Mamouni,
Université Moulay Ismail, Faculté des Sciences et Techniques Département de Mathématiques, BP. 509-Boutalamine 52000 Errachidia, Maroc
• Lahcen Oukhtite,
Université Moulay Ismail, Faculté des Sciences et Techniques Département de Mathématiques, BP. 509-Boutalamine 52000 Errachidia, Maroc
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## Abstract

Let $R$ be a ring, $g$ an endomorphism of $R$. The additive mapping $d\colon R\rightarrow R$ is called Jordan semiderivation of $R$, associated with $g$, if $$d(x^2)=d(x)x+g(x)d(x)=d(x)g(x)+xd(x)\quad \text{and}\quad d(g(x))=g(d(x))$$ for all $x\in R$. The additive mapping $F\colon R\rightarrow R$ is called generalized Jordan semiderivation of $R$, related to the Jordan semiderivation $d$ and endomorphism $g$, if $$F(x^2)=F(x)x+g(x)d(x)=F(x)g(x)+xd(x)\quad \text{and}\quad F(g(x))=g(F(x))$$ for all $x\in R$. In the present 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: semiderivation, generalized semiderivation, Jordan semiderivation, prime ring
 MSC Classifications: 16W25 - Derivations, actions of Lie algebras

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