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 formula_2:
Basic definitions and properties.
For a ring "R" and ideal "I", multiplication in formula_4 is defined as follows: First, consider homogeneous elements formula_5 and formula_6 and suppose formula_7 is a representative of "a" and formula_8 is a representative of "b". Then define formula_9 to be the equivalence class of formula_10 in formula_11. Note that this is well-defined modulo formula_12. 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 formula_13, the initial form of "f" in formula_14, written formula_15, is the equivalence class of "f" in formula_16 where "m" is the maximum integer such that formula_17. If formula_18 for every "m", then set formula_19. The initial form map is only a map of sets and generally not a homomorphism. For a submodule formula_20, formula_21 is defined to be the submodule of formula_14 generated by formula_23. This may not be the same as the submodule of formula_24 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 formula_2 is an integral domain, then "R" is itself an integral domain.
gr of a quotient module.
Let formula_20 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:
where
called the "submodule generated by the initial forms of the elements of formula_30.
Examples.
Let "U" be the universal enveloping algebra of a Lie algebra formula_31 over a field "k"; it is filtered by degree. The Poincaré–Birkhoff–Witt theorem implies that formula_32 is a polynomial ring; in fact, it is the coordinate ring formula_33.
The associated graded algebra of a Clifford algebra is an exterior algebra; i.e., a Clifford algebra degenerates to an exterior algebra.
Generalization to multiplicative filtrations.
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 formula_35. The graded ring associated with this filtration is formula_36. Multiplication and the initial form map are defined as above.