Commutative Noetherian Domains of Krull Dimension One k is an alegraically closed field and A is a commutative k-algebra. We also know that A is a Noetherian domain and its Krull dimension is one. Are there any necessary and sufficient conditions on A under which A becomes finitely generated module over a polynomial algebra k[c] for some c in A? Does anybody know any papers or books that discuss this? 
Thanks guys.  
 A: Dear Amitsur, 
It might help you to think geometrically.  For example, $k[x,x^{-1}]$ is the ring of functions
on a hyperbola $xy = 1$, and the projection from this hyperbola to the line $x = y$ is a finite projection.  This corresponds to the fact that $k[x,x^{-1}]$ is finitely generated as a module over $k[x  + x^{-1}].$ (If we write $f = x + x^{-1}$, then $x^2 - f x +1 = 0$
and $x^{-2} - f x^{-1} + 1 = 0$.)
A: This is always the case, by Noether normalization. For a proof, see for examlple, Eisenbud - "Commutative algebra with a view towards algebraic geometry" - Theorem 13.3.
Edit: this is false, one need a finiteness assumption. See comments below.
A: This follows from a direct generalization of the Noether normalization lemma.  It is covered in these notes from Mel Hochster.  These notes prove it in a pretty general form (when the base ring is only an integral domain rather than a field).
Edit: A sufficient condition is that the algebra is finitely generated, but it is clearly not necessary.
Edit 2: I misread the question.  I thought he was asking if A is finitely generated over some polynomial algebra (including infinitely generated polynomial algebras).
