In her 1926 paper on invariant theory, Emmy Noether uses a certain "finiteness criterion" which I wish to translate and maybe find a more modern reference to. The original wording is:
Ein Integritätsbereich $\mathfrak{J}$ aus Polynomen mit Koeffizienten aus einem Körper, ist dann und nur dann ein endlicher, wenn in $\mathfrak{J}$ ein endlicher Unterring $\mathfrak{R}$ enthalten ist derart, daß $\mathfrak{J}$ algebraisch und ganz von $\mathfrak{R}$ abhängt.
My first shot at translating this was:
Let $A\subseteq k[x_1,\dots,x_n]$ be a $k$-subalgebra. Then $A$ is finitely generated if and only if there exists a finitely generated $k$-subalgebra $R\subseteq A$ such that $A$ is integral over $R$.
I translated "endlicher Integritätsbereich" to "finitely generated $k$-algebra which is an integral domain", but the second condition is fulfilled anyway. I'm not 100% sure about the notion "Integritätsbereich" in such an older paper though. Also, I translated "endlicher Unterring" to "finitely generated subalgebra", although "endlich" strictly speaking means "finite". Is the translation correct as it stands?
Do you know of any reference or modern proof of this result?
