The UFD field lemma

This page contains a result which it refers to as the UFD field lemma. I was wondering if anybody knew of any other references which discuss this result--this page is the only place I've seen it.

The UFD field lemma appears to assert that if $R$ is a unique factorization domain containing infinitely many prime elements, if $F$ is the field of fractions of $R$, and if $A$ is a finitely generated $R$-algebra which is a field and is algebraic over $F$, then $A$ does not contain $F$.

I'm looking for other sources because I find the exposition on that page a little hard to follow.

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See Proposition 7.8 in Atiyah-Macdonald, where in their notation, A=R, B=F, and C=A (your A). – user641 Sep 12 '10 at 10:03
If you are interested in the Nullstellensatz you might like to see Dan Bernstein's take on it at cr.yp.to/zgk.html . – Robin Chapman Sep 12 '10 at 10:49
Thanks very much for the references! – Zach Conn Sep 13 '10 at 7:39