Noether's normalisation lemma says that if $R$ is an integral domain, finitely generated over a field $k$, with transcendence degree $n$ over $k$, then there exist elements $x\_{1}, x\_{2}, \ldots x\_{n} \in R$, algebraically independent over $k$, such that $R$ is integally dependent on the subring $k[x\_{1}, \ldots, x\_{n}]$ generated by the $x$'s. 


I am interested in knowing if there is a strengthened version of this where in addition we can arrange for $R$ to be separable over $k[x\_{1}, x\_{2}, \ldots x\_{n}]$. Looking at the proof which appears in Mumford's "Introduction to Algebraic Geometry", it seems as though that proof can be modified to prove this stronger result. Is this something which has been noted in the literature before? A reference would be very helpful.



A strengthened version of Noether's normalisation lemma?
Yes [when, as noted by Laurent Moret-Bailly, $\mathrm{Frac}(R)$ is separable over $k$]. See David Eisenbud 'Commutative Algebra With a View Toward Algebraic Geometry' Corollary 16.18 about this.


[Edit:] Here is the theorem: if $R$ is an integral domain, finitely generated over a field $k$, with transcendence degree $n$ over $k$, and assume that $\mathrm{Frac}(R)$ is separable over $k$, then there exist elements $x\_1,x\_2,\ldots x\_n\in R$, algebraically independent over $k$, such that $R$ is integrally dependent on the subring $k[x\_1,\ldots ,x\_n]$, and $R$ is separable over $k[x\_1,\ldots ,x\_n]$. 


