On some additive mappings on division rings

Document Type : Research Paper


1 Department of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh, India

2 Department of Mathematics Faculty of Science Cairo University Giza 12613, Egypt



 Let $D$ be a division ring such that char$(D) \neq 2$  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 $f(x)\alpha(x^{-1}) + \beta(x)g(x^{-1}) = 0$ for all $0 \neq x\in D.$ 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{C1} and \cite{C2}, respectively.