Hilbert series and dimension for a graded ring (and integrable systems)
This question isn't important for algebra, I guess, but for the integrable systems with polynomial integrals.
Consider $R$ --- graded ring (actually subring of some $K[x_1, ..., x_n]$, generated by a set of polynomials) over $\mathbb C$. For a finitely-generated ring I know, that its dimention (degree of transendence) is uniqely determined by its Hilbert series. Is it true for infinitely-generated rings of such nature?
In particular I am interested in secific case, when there are two graded rings with same Hilbert series, one of which is finitely generated and other is not. Is it true that they will have the same degree of transendence?