On additive mappings in rings with identity element

Abstract

Let R be an associative $(n+1)!$-torsion free ring with identity and n be a fixed positive integer. Suppose that D, T, F are additive mappings on R. We prove that (a) let R be a semi prime ring satisfying (1) $2D(x^{n+1}) = D(x^n)x+x^nD(x)+D(x)x^n+xD(x^n)$ for all $x \in R$, then D is a derivation; (2) if $4T(x^{n+1})=T(x^n)x+x^nT(x)+T(x)x^n+xT(x^n)$ for all $x \in R,$ then there exists $a\in Z(R)$ such that $T(x)= ax$ for all $x \in R$; (b) if $2F(x^{n+1})=F(x)x^n+xD(x^n)+F(x^n)x+x^nD(x)$ for all $x \in R,$ then D is a Jordan derivation and F is a Jordan generalized derivation.