Given a commutative unit ring R and a filtration F:...⊆I_2 ⊆I_1 ⊆I_0 = R of ideals of R, the associated graded ring of R with respect to F is the graded ring gr_F(R) = I_0/I_1 ⊕I_1/I_2 ⊕I_2/I_3 ⊕.... The addition is defined componentwise, and the product is defined as follows. If a = [α]_i element I_i/I_(i + 1) is the residue class of α element I_i mod I_(i + 1), and b = [β]_i element I_j/I_(j + 1) is the residue class of β element I_j mod I_(j + 1), then a·b = [αβ]_(i + j) is the residue class of αβ element I_(i + j) mod I_(i + j + 1).

associated graded module | Hilbert-Samuel function | Rees ring