Identities with inverses on matrix rings

Motivated by [1, 2], the goal of the paper is to study certain identities with inverses on matrix rings. Given D a division ring, we characterize additive maps f, g: D ? D satisfying the identity f(x)x?1 + xg(x?1) = 0 for all invertible x ? D. Let R be a matrix ring over a division ring of characteristic not 2. We also characterize additive maps f, g: R ? R satisfying the identity f(x)x?1 + xg(x?1) = 0 for all invertible x ? R. Precisely, there exist an element q ? R and a derivation d of R such that f(x) = xq + d(x) and g(x) = ?qx + d(x) for all x ? R. This affirmatively answers the question below Theorem 4 in [1] due to L. Catalano. © 2019, © 2019 Informa UK Limited, trading as Taylor & Francis Group.