Generalization of Herstein theorem and its applications to range inclusion problems

Let R   be an associative ring. An additive mapping d:R→Rd:R→R is called a Jordan derivation if d(x2)=d(x)x+xd(x)d(x2)=d(x)x+xd(x) holds for all x∈Rx∈R. The objective of the present paper is to characterize a prime ring R which admits Jordan derivations d and g   such that [d(xm),g(yn)]=0[d(xm),g(yn)]=0 for all x,y∈Rx,y∈R or d(xm)∘g(yn)=0d(xm)∘g(yn)=0 for all x,y∈Rx,y∈R, where m⩾1m⩾1 and n⩾1n⩾1 are some fixed integers. This partially extended Herstein’s result in [6, Theorem 2], to the case of (semi)prime ring involving pair of Jordan derivations. Finally, we apply these purely algebraic results to obtain a range inclusion result of continuous linear Jordan derivations on Banach algebras.