On some additive mappings on division rings

Let D be a division ring such that char(D) \neq ۲  and \alpha,\beta:D\rightarrow D be automorphisms of D. The main purpose of this paper is to characterizes additive maps                              f and g satisfying the identity f(x)\alpha(x^{-۱}) + \beta(x)g(x^{-۱}) = ۰ for all ۰ \neq x\in D. As an application, we describe the structure of an additive map f satisfying the identity                            f(x)\alpha(y)+\beta(x)f(y) =l for all x,y\in D such that xy=a, where l,a\in D and a is nonzero. With this, many known results can be either generalized or deduced. In particular, we generalized the results proved in\cite{C۱} and \cite{C۲}, respectively.

Keywords:

Division ring , Derivation , (alpha , beta)-derivation , functional identity