Abstract view
Generalized Jordan Semiderivations in Prime Rings


Published:20150206
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. 509Boutalamine 52000 Errachidia, Maroc
Lahcen Oukhtite,
Université Moulay Ismail, Faculté des Sciences et Techniques Département de Mathématiques, BP. 509Boutalamine 52000 Errachidia, Maroc
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 $F=d$.