Associated graded ring

In mathematics, the associated graded ring of a ring R with respect to a proper ideal I is the graded ring: Similarly, if M is a left R-module, then the associated graded module is the graded module over

: For a ring R and ideal I, multiplication in

is defined as follows: First, consider homogeneous elements

Then define

{\displaystyle ab}

to be the equivalence class of

Note that this is well-defined modulo

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

, the initial form of f in

, is the equivalence class of f in

where m is the maximum integer such that

The initial form map is only a map of sets and generally not a homomorphism.

For a submodule

is defined to be the submodule of

generated by

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

is an integral domain, then R is itself an integral domain.

be left modules over a ring R and I an ideal of R. Since (the last equality is by modular law), there is a canonical identification:[2] where called the submodule generated by the initial forms of the elements of

Let U be the universal enveloping algebra of a Lie algebra

over a field k; it is filtered by degree.

The Poincaré–Birkhoff–Witt theorem implies that

is a polynomial ring; in fact, it is the coordinate ring

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 such that

The graded ring associated with this filtration is

Multiplication and the initial form map are defined as above.