In mathematics, the associated graded ring of a ring R with respect to a proper ideal I is the graded ring:
gr I R = ⊕ n = 0 ∞ I n / I n + 1 .
Similarly, if M is a left R-module, then the associated graded module is the graded module over gr I R :
gr I M = ⊕ 0 ∞ I n M / I n + 1 M .
Basic definitions and properties
For a ring R and ideal I, multiplication in gr I R is defined as follows: First, consider homogeneous elements a ∈ I i / I i + 1 and b ∈ I j / I j + 1 and suppose a ′ ∈ I i is a representative of a and b ′ ∈ I j is a representative of b. Then define a b to be the equivalence class of a ′ b ′ in I i + j / I i + j + 1 . Note that this is well-defined modulo I i + j + 1 . Multiplication of inhomogeneous elements is defined by using the distributive property.
A ring or module may be related to its associated graded ring or module through the initial form map. Let M be an R-module and I an ideal of R. Given f ∈ M , the initial form of f in gr I M , written i n ( f ) , is the equivalence class of f in I m M / I m + 1 M where m is the maximum integer such that f ∈ I m M . If f ∈ I m M for every m, then set i n ( f ) = 0 . The initial form map is only a map of sets and generally not a homomorphism. For a submodule N ⊂ M , i n ( N ) is defined to be the submodule of gr I M generated by { i n ( f ) | f ∈ N } . This may not be the same as the submodule of gr I M generated by the only initial forms of the generators of N.
A ring inherits some "good" properties from its associated graded ring. For example, if R is a noetherian local ring, and gr I R is an integral domain, then R is itself an integral domain.
Let U be the enveloping algebra of a Lie algebra g over a field k; it is filtered by degree. The Poincaré–Birkhoff–Witt theorem implies that gr U is a polynomial ring; in fact, it is the coordinate ring k [ g ∗ ] .
The associated graded algebra of a Clifford algebra is an exterior algebra; i.e., a Clifford algebra degenerates to an exterior algebra.
The associated graded can also be defined more generally for multiplicative descending filtrations of R (see also filtered ring.) Let F be a descending chain of ideals of the form
R = I 0 ⊃ I 1 ⊃ I 2 ⊃ ⋯ such that I j I k ⊂ I j + k . The graded ring associated with this filtration is gr F R = ⊕ n = 0 ∞ I n / I n + 1 . Multiplication and the initial form map are defined as above.
