Noether's normalization in skew polynomial rings

We study Noether's normalization lemma for finitely generated algebras over a division algebra. In its classical form, the lemma states that if I is a proper ideal of the ring R=F[t₁,…,tₙ] of polynomials over a field F, then the quotient ring $R/I$ is a finite extension of a polynomial ring over F. We prove that the lemma holds when R=D[t₁,…,tₙ] is the ring of polynomials in n central variables over a division algebra D. We provide examples demonstrating that Noether's normalization may fail for the skew polynomial ring D[t₁,…,tₙ;σ₁,…,σₙ] with respect to commuting automorphisms σ₁,…,σₙ of D. We give a sufficient condition for σ₁,…,σₙ under which the normalization lemma holds for such ring. In the case where $D=F$ is a field, this sufficient condition is proved to be necessary.