ON SOME ADDITIVE MAPPINGS ON DIVISION RINGS

Let d be a division ring such that char(d) 6= 2 and α, β : d → d be automorphisms of d. the main purpose of this paper is to characterizes additive maps f and g satisfying the identity f(x)α(x−1 ) + β(x)g(x−1 ) = 0 for all 0 6= x ∈ d. as an application, we describe the structure of an additive map f satisfying the identity f(x)α(y) + β(x)f(y) = l for all x, y ∈ d such that xy = a, where l, a ∈ d and a is nonzero. with this, many known results can be either generalized or deduced. in particular, we generalized the results proved in [2] and [3], respectively.