Artinian ring
Suppose "M" is a finitely generated graded module over formula_14 with an Artinian ring (e.g., a field) "A". Then the Poincaré series of "M" is a polynomial with integral coefficients divided by formula_15. The standard proof today is an induction on "n". Hilbert's original proof made a use of Hilbert's syzygy theorem (a projective resolution of "M"), which gives more homological information.