Noether's normalization in skew polynomial rings

Abstract

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 $R=F[t_1,\dots ,t_n]$ 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 $R=D[t_1,\dots ,t_n]$ 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 $D[t_1,\dots ,t_n;\ddot{I}{ƒ}_1,\dots ,\ddot{I}{ƒ}_n]$ with respect to commuting automorphisms $\ddot{I}{ƒ}_1,\dots ,\ddot{I}{ƒ}_n$ of $D$. We give a sufficient condition for $\ddot{I}{ƒ}_1,\dots ,\ddot{I}{ƒ}_n$ 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.