On a certain functional identity involving inverses in division rings and local rings

Let Formula: see text be a division ring, Formula: see text an automorphism of Formula: see text and Formula: see text a fixed element. We determine the forms of additive maps Formula: see text satisfying the identity Formula: see text for every nonzero Formula: see text. We show that Formula: see text and Formula: see text are both generalized Formula: see text-derivations of Formula: see text except possibly when Formula: see text is the identity automorphism and Formula: see text is an imperfect field of characteristic Formula: see text. We further study the same rational identity in the context of local rings under some mild assumptions.