 Let be a graded algebra and a homogeneous element of degree in which is not a zero divisor.Then we have 
It follows from the additivity on the exact sequence 
where the arrow labeled is the multiplication by , and formula_33 is the graded module which is obtained from by shifting the degrees by , in order that the multiplication by has degree 0.This implies that formula_34

The Hilbert series of the polynomial ring formula_35 in formula_36 indeterminates is 
It follows that the Hilbert polynomial is 

The proof that the Hilbert series has this simple form is obtained by applying recursively the previous formula for the quotient by a non zero divisor (here formula_39) and remarking that formula_40

A graded algebra generated by homogeneous elements of degree 1 has Krull dimension zero if the maximal homogeneous ideal, that is the ideal generated by the homogeneous elements of degree 1, is nilpotent.