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 is an element of 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 is an element of R. Precisely, there exist an element q is an element of R and a derivation d of R such that f(x)=xq+d(x) and g(x)=-qx+d(x) for all x is an element of R. This affirmatively answers the question below Theorem 4 in [1] due to L. Catalano.