k is an alegraically closed field and A is a commutative kalgebra. 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.

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 $x^{2}  f x^{1} + 1 = 0$.) 


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. 


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: 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). 

