The Hilbert series of R/I for I a polynomial ideal, as the generating function of the Hilbert function.
The elements of R/I_X, while no longer well-defined as polynomials, are still well-defined as functions on X.
Given S a subspace of R, when is there a well-defined multiplication on R/S? Answer: S an ideal.
(Actually, we only showed that if S is an ideal, then the multiplication on R/S is well-defined. But necessity is easy: the zero element of R/S is the coset 0 + S = S itself, and we want C*0 = 0 for any element C in R/S. Upstairs in R, that says r*s in S for any r in any C, any s in S.)
