Given a module M over a commutative unit ring R and a filtration F:...⊆I_2 ⊆I_1 ⊆I_0 = R of ideals of R, the associated graded module of M with respect to F is gr_F(M) = I_0 M/I_1 M⊕I_1 M/I_2 M⊕I_2 M/I_3 M⊕..., which is a graded module over the associated graded ring gr_F(R) with respect to the addition and the multiplication by scalars defined componentwise. If I is a proper ideal of R, then the notation gr_I(M) indicates the associated graded module of M with respect to the I-adic filtration of R, gr_I(M) = M/I M⊕I M/I^2 M⊕I^2 M/I^3 M⊕....

associated graded ring | Hilbert-Samuel function | Rees module