Extended Jacobson density theorem for rings with skew derivations

Let A be a ring with a simple right module M and D=End(MA). Let {θ₁, θ₂, θ_n} be a reduced set of skew derivations, where each i is a i-derivation of A satisfying ij=ji and ij=ji for all i, j. Let 1, m be distinct words of the form [image omitted], where each isp in the case of char(D)=p0. Let 1, be mutually M-independent automorphisms of A. Then for any D-independent elements x1, x2, xkM and any elements zijtM, there exists aA such that zijt=xiajt for all i=1, 2, k, j=1, 2, m, t=1, .