# A criterion for finite generation of subalgebras of a polynomial ring

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?

• @TTS: It may be my translation is wrong and some of the notions are thus not what is originally stated. If $A$ is finitely generated, of course you can take $R=A$. Why is the other direction easy though? Maybe "endlicher Unterring" really means "finite subring", but how Noether uses it I don't think that's the case. Jun 27, 2013 at 15:17
• Could 'endlicher Unterring' mean subring of finite index?
– HSN
Jun 27, 2013 at 16:17
• @Randal'Thor Do you think this solution is related/what you need? Jun 27, 2013 at 16:36
• @Randal'Thor Also, there is an inexpensive, albeit physics oriented, book called Emmy Noether's Wonderful theorem. Even if it doesn't have the mathematics you are looking for, I'm sure it will have the references you want (especially good translations.) Jun 27, 2013 at 16:40
• @rschwieb Thank you for the suggestion and the link to the other thread. I will look into this very soon! Jul 2, 2013 at 16:28