Generalized Jordan Semiderivations in Prime Rings

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 $F\,=\,d$ .